Daily Papers

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its theoretical understanding has lagged behind practice. In particular, practical implementations incorporate Nesterov momentum, compute the polar factor only approximately, and operate with stochastic gradients that may be heavy-tailed. We close this gap by developing a convergence theory for Muon with Nesterov momentum and inexact polar decomposition in non-convex matrix optimization under heavy-tailed noise. Our analysis builds on a unified framework for inexact polar decomposition that captures practical iterative approximations such as Newton-Schulz and quantifies how their errors propagate through the optimization dynamics. Under this framework, we establish an optimal iteration and sample complexity of O left(varepsilon^{-(3α-2){(α-1)}} right) for finding an varepsilon-stationary point, where αin(1,2] denotes the heavy-tail index. For the inexact-polar setting with σ_1=0, we also provide guarantees that do not require prior knowledge of α. We analyze a randomized low-rank polar decomposition that is substantially more efficient than full-space methods while remaining compatible with our theory. Numerical experiments further demonstrate the effectiveness of the proposed inexact and randomized variants.

An increasing number of studies have focused on stochastic first-order methods (SFOMs) under heavy-tailed gradient noises, which have been observed in the training of practical deep learning models. In this paper, we focus on two types of gradient noises: one is sub-Weibull noise, and the other is noise under the assumption that it has a bounded p-th central moment (p-BCM) with pin (1, 2]. The latter is more challenging due to the occurrence of infinite variance when pin (1, 2). Under these two gradient noise assumptions, the in-expectation and high-probability convergence of SFOMs have been extensively studied in the contexts of convex optimization and standard smooth optimization. However, for weakly convex objectives-a class that includes all Lipschitz-continuous convex objectives and smooth objectives-our understanding of the in-expectation and high-probability convergence of SFOMs under these two types of noises remains incomplete. We investigate the high-probability convergence of the vanilla stochastic subgradient descent (SsGD) method under sub-Weibull noises, as well as the high-probability and in-expectation convergence of clipped SsGD under the p-BCM noises. Both analyses are conducted in the context of weakly convex optimization. For weakly convex objectives that may be non-convex and non-smooth, our results demonstrate that the theoretical dependence of vanilla SsGD on the failure probability and number of iterations under sub-Weibull noises does not degrade compared to the case of smooth objectives. Under p-BCM noises, our findings indicate that the non-smoothness and non-convexity of weakly convex objectives do not impact the theoretical dependence of clipped SGD on the failure probability relative to the smooth case; however, the sample complexity we derived is worse than a well-known lower bound for smooth optimization.

Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process. To manage this realistic and challenging setting, new mechanisms, such as gradient clipping and gradient normalization, have been introduced to ensure the convergence of first-order algorithms. However, adaptive gradient methods, a famous class of modern optimizers that includes popular Adam and AdamW, often perform well even without any extra operations mentioned above. It is therefore natural to ask whether adaptive gradient methods can converge under heavy-tailed noise without any algorithmic changes. In this work, we take the first step toward answering this question by investigating a special case, AdaGrad, the origin of adaptive gradient methods. We provide the first provable convergence rate for AdaGrad in non-convex optimization when the tail index p satisfies 4/3<pleq2. Notably, this result is achieved without requiring any prior knowledge of p and is hence adaptive to the tail index. In addition, we develop an algorithm-dependent lower bound, suggesting that the existing minimax rate for heavy-tailed optimization is not attainable by AdaGrad. Lastly, we consider AdaGrad-Norm, a popular variant of AdaGrad in theoretical studies, and show an improved rate that holds for any 1<pleq2 under an extra mild assumption.

We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered Lévy noise process with finite p-th order moments, 1 < p leq 2. This scheme -- which we call the Lévy mirror flow (LMF) -- arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when p < 2 -- the heavy noise regime -- the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains ε-optimality within O(ε^{-p/(p-1)}) time in the convex case, and within tilde O(ε^{-1/(p-1)}) time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.

Methods with adaptive stepsizes, such as AdaGrad and Adam, are essential for training modern Deep Learning models, especially Large Language Models. Typically, the noise in the stochastic gradients is heavy-tailed for the later ones. Gradient clipping provably helps to achieve good high-probability convergence for such noises. However, despite the similarity between AdaGrad/Adam and Clip-SGD, the high-probability convergence of AdaGrad/Adam has not been studied in this case. In this work, we prove that AdaGrad (and its delayed version) can have provably bad high-probability convergence if the noise is heavy-tailed. To fix this issue, we propose a new version of AdaGrad called Clip-RAdaGradD (Clipped Reweighted AdaGrad with Delay) and prove its high-probability convergence bounds with polylogarithmic dependence on the confidence level for smooth convex/non-convex stochastic optimization with heavy-tailed noise. Our empirical evaluations, including NLP model fine-tuning, highlight the superiority of clipped versions of AdaGrad/Adam in handling the heavy-tailed noise.

Heavy-tailed stochastic gradient noise, commonly observed in transformer models, can destabilize the optimization process. Recent works mainly focus on developing and understanding approaches to address heavy-tailed noise in the centralized or distributed, synchronous setting, leaving the interactions between such noise and asynchronous optimization underexplored. In this work, we investigate two communication schemes that handle stragglers with asynchronous updates in the presence of heavy-tailed gradient noise. We propose and theoretically analyze algorithmic modifications based on delay-aware learning rate scheduling and delay compensation to enhance the performance of asynchronous algorithms. Our convergence guarantees under heavy-tailed noise match the rate of the synchronous counterparts and improve delay tolerance compared with existing asynchronous approaches. Empirically, our approaches outperform prior synchronous and asynchronous methods in terms of accuracy/runtime trade-offs and are more robust to hyperparameters in both image and language tasks.

Autonomous systems often must predict the motions of nearby agents from partial and noisy data. This paper asks and answers the question: "can we learn, in real-time, a nonlinear predictive model of another agent's motions?" Our online framework denoises and forecasts such dynamics using a modified sliding-window Hankel Dynamic Mode Decomposition (Hankel-DMD). Partial noisy measurements are embedded into a Hankel matrix, while an associated Page matrix enables singular-value hard thresholding (SVHT) to estimate the effective rank. A Cadzow projection enforces structured low-rank consistency, yielding a denoised trajectory and local noise variance estimates. From this representation, a time-varying Hankel-DMD lifted linear predictor is constructed for multi-step forecasts. The residual analysis provides variance-tracking signals that can support downstream estimators and risk-aware planning. We validate the approach in simulation under Gaussian and heavy-tailed noise, and experimentally on a dynamic crane testbed. Results show that the method achieves stable variance-aware denoising and short-horizon prediction suitable for integration into real-time control frameworks.

In large-scale optimization, the cheapness and effectiveness of update steps are the most crucial factors for a successful optimizer. Sign-based optimizers like Lion or Signum produce cheap per-step updates, whereas Muon's spectral matrix-sign update gives a much stronger direction at a substantially higher per-step cost. In this work, we propose LionMuon, which retains the effectiveness of Muon steps while considerably cutting the averaged iteration cost, similar to sign-based methods. It alternates between Lion's and Muon's updates on a fixed period P, sharing a single dual-EMA momentum buffer between them. The optimizer state memory therefore matches Lion and is exactly half of AdamW's. A simpler single-EMA variant, SignMuon, by itself already outperforms pure Muon. At P = 2, LionMuon Pareto-dominates Muon, Lion, Signum, and AdamW on every dataset and architecture we tested at 124M model size, reaching lower validation loss at lower compute, and the same advantage persists at 355M and 720M scale. On the theory side, we prove sharp complexity bounds under heavy-tailed noise which are governed by period-averaged smoothness and noise that interpolate between Muon's and Lion's constants. These bounds predict the compute-optimal period and the conditions under which LionMuon outruns Muon and Lion. Code: https://github.com/brain-lab-research/lion-muon

We study the problem of robustly learning Gaussian Single Index Models (SIMs) in the presence of heavy-tailed noise and a constant fraction of adversarially corrupted covariates and responses. Prior work on robust recovery has considered settings such as linear regression (Pensia et al., JASA 2024), strictly monotonic link functions (Awasthi et al., NeurIPS 2022), and phase retrieval (Buna and Rebeschini, AISTATS 2025). However, these techniques do not extend to generic asymmetric non-monotonic link functions such as GeLU and Swish, which arise naturally as scalar primitives in modern gated neural architectures. We close this gap by giving the first robust recovery algorithm with near-linear sample and time complexity for generic non-monotonic link functions, thereby establishing the first robust recovery guarantees for a broad family of nonlinear SIMs for which no guarantees were previously known. Our central contribution is a new structural understanding of the Gaussian squared-loss landscape under adversarial contamination. Crucially, we prove that for a broad class of nonlinear non-monotonic SIMs, a dimension-independent, constant-radius convex basin exists around the ground truth and is efficiently reachable via robust spectral initialization even under adversarial contamination. Prior works fail to establish both guarantees simultaneously, thereby either breaking down under adversarial contamination or failing to handle generic non-monotonic link functions. Together, these structural insights yield a principled warm start for robust gradient descent that provably converges to a final estimation error of O(σsqrtε) in O(nd) time with O(d) samples, where ε is the contamination fraction.

We study the dynamics of Stochastic Gradient Descent (SGD), which is known to steer deep neural networks toward invariant sets that correspond to simpler subnetworks. How this steering unfolds over time remains poorly understood. We answer this by modeling the stochastic gradient flow (SGF) as a percolation process, in which architectural symmetries force subnetworks to merge in discrete simultaneous blocks rather than one at a time. These structural transitions register as variance spikes in a macroscopic order parameter, echoing physical phase transitions. We further show this trapping mechanism and its associated scaling cascade extend to Adam and AdamW under an explicit heavy-tailed noise model.

In the past several years, the last-iterate convergence of the Stochastic Gradient Descent (SGD) algorithm has triggered people's interest due to its good performance in practice but lack of theoretical understanding. For Lipschitz convex functions, different works have established the optimal O(log(1/delta)log T/T) or O(log(1/delta)/T) high-probability convergence rates for the final iterate, where T is the time horizon and delta is the failure probability. However, to prove these bounds, all the existing works are either limited to compact domains or require almost surely bounded noises. It is natural to ask whether the last iterate of SGD can still guarantee the optimal convergence rate but without these two restrictive assumptions. Besides this important question, there are still lots of theoretical problems lacking an answer. For example, compared with the last-iterate convergence of SGD for non-smooth problems, only few results for smooth optimization have yet been developed. Additionally, the existing results are all limited to a non-composite objective and the standard Euclidean norm. It still remains unclear whether the last-iterate convergence can be provably extended to wider composite optimization and non-Euclidean norms. In this work, to address the issues mentioned above, we revisit the last-iterate convergence of stochastic gradient methods and provide the first unified way to prove the convergence rates both in expectation and in high probability to accommodate general domains, composite objectives, non-Euclidean norms, Lipschitz conditions, smoothness, and (strong) convexity simultaneously. Additionally, we extend our analysis to obtain the last-iterate convergence under heavy-tailed noises.

There are several applications of stochastic optimization where one can benefit from a robust estimate of the gradient. For example, domains such as distributed learning with corrupted nodes, the presence of large outliers in the training data, learning under privacy constraints, or even heavy-tailed noise due to the dynamics of the algorithm itself. Here we study SGD with robust gradient estimators based on estimating the median. We first consider computing the median gradient across samples, and show that the resulting method can converge even under heavy-tailed, state-dependent noise. We then derive iterative methods based on the stochastic proximal point method for computing the geometric median and generalizations thereof. Finally we propose an algorithm estimating the median gradient across iterations, and find that several well known methods - in particular different forms of clipping - are particular cases of this framework.

Most existing video anomaly detectors rely solely on RGB frames, which lack the temporal resolution needed to capture abrupt or transient motion cues, key indicators of anomalous events. To address this limitation, we propose Image-Event Fusion for Video Anomaly Detection (IEF-VAD), a framework that synthesizes event representations directly from RGB videos and fuses them with image features through a principled, uncertainty-aware process. The system (i) models heavy-tailed sensor noise with a Student`s-t likelihood, deriving value-level inverse-variance weights via a Laplace approximation; (ii) applies Kalman-style frame-wise updates to balance modalities over time; and (iii) iteratively refines the fused latent state to erase residual cross-modal noise. Without any dedicated event sensor or frame-level labels, IEF-VAD sets a new state of the art across multiple real-world anomaly detection benchmarks. These findings highlight the utility of synthetic event representations in emphasizing motion cues that are often underrepresented in RGB frames, enabling accurate and robust video understanding across diverse applications without requiring dedicated event sensors. Code and models are available at https://github.com/EavnJeong/IEF-VAD.

Muon has recently shown promising results in LLM training. In this work, we study how to further improve Muon. We argue that Muon's orthogonalized update rule suppresses the emergence of heavy-tailed weight spectra and over-emphasizes the training along noise-dominated directions. Motivated by the Heavy-Tailed Self-Regularization (HT-SR) theory, we propose HTMuon. HTMuon preserves Muon's ability to capture parameter interdependencies while producing heavier-tailed updates and inducing heavier-tailed weight spectra. Experiments on LLM pretraining and image classification show that HTMuon consistently improves performance over state-of-the-art baselines and can also serve as a plug-in on top of existing Muon variants. For example, on LLaMA pretraining on the C4 dataset, HTMuon reduces perplexity by up to 0.98 compared to Muon. We further theoretically show that HTMuon corresponds to steepest descent under the Schatten-q norm constraint and provide convergence analysis in smooth non-convex settings. The implementation of HTMuon is available at https://github.com/TDCSZ327/HTmuon.

We introduce a novel framework for efficient sampling from complex, unnormalised target distributions by exploiting multiscale dynamics. Traditional score-based sampling methods either rely on learned approximations of the score function or involve computationally expensive nested Markov chain Monte Carlo (MCMC) loops. In contrast, the proposed approach leverages stochastic averaging within a slow-fast system of stochastic differential equations (SDEs) to estimate intermediate scores along a diffusion path without training or inner-loop MCMC. Two algorithms are developed under this framework: MultALMC, which uses multiscale annealed Langevin dynamics, and MultCDiff, based on multiscale controlled diffusions for the reverse-time Ornstein-Uhlenbeck process. Both overdamped and underdamped variants are considered, with theoretical guarantees of convergence to the desired diffusion path. The framework is extended to handle heavy-tailed target distributions using Student's t-based noise models and tailored fast-process dynamics. Empirical results across synthetic and real-world benchmarks, including multimodal and high-dimensional distributions, demonstrate that the proposed methods are competitive with existing samplers in terms of accuracy and efficiency, without the need for learned models.

Score-based diffusion models generate samples from an unknown target distribution using a time-reversed diffusion process. While such models represent state-of-the-art approaches in industrial applications such as artificial image generation, it has recently been noted that their performance can be further improved by considering injection noise with heavy tailed characteristics. Here, I present a generalization of generative diffusion processes to a wide class of non-Gaussian noise processes. I consider forward processes driven by standard Gaussian noise with super-imposed Poisson jumps representing a finite activity Levy process. The generative process is shown to be governed by a generalized score function that depends on the jump amplitude distribution and can be estimated by minimizing a simple MSE loss as in conventional Gaussian models. Both probability flow ODE and SDE formulations are derived using basic technical effort. A detailed implementation for a pure jump process with Laplace distributed amplitudes yields a generalized score function in closed analytical form and is shown to outperform the equivalent Gaussian model in specific parameter regimes.

In modern machine learning, parallelization of training is an important strategy for increasing scale. Asynchronous stochastic gradient descent (ASGD), which maximizes the utilization of available hardware by avoiding waiting for slow workers. However, with constant step sizes, the convergence of ASGD is nonetheless affected negatively by slow workers due to large delays in updates. At the same time, it has been empirically observed in asynchronous training of deep learning models that gradient clipping "stabilizes" training. In this work, we provide a theoretical justification for this behavior, as we show that clipping removes the dependence of the maximum delay in the oracle complexity. We employ a sub-Weibull model of gradient noise which generalizes sub-Gaussian and sub-exponential distributions to more heavy-tailed distributions, motivated by empirical observations in deep learning. We show convergence in expectation, and the first time in asynchronous optimization, convergence with high probability.

Training LLMs at ultra-low precision remains a formidable challenge. Direct low-bit QAT often suffers from convergence instability and substantial training costs, exacerbated by quantization noise from heavy-tailed outlier channels and error accumulation across layers. To address these issues, we present Bit-by-Bit, a progressive QAT framework with outlier channel splitting. Our approach integrates three key components: (1) block-wise progressive training that reduces precision stage by stage, ensuring stable initialization for low-bit optimization; (2) nested structure of integer quantization grids to enable a "train once, deploy any precision" paradigm, allowing a single model to support multiple bit-widths without retraining; (3) rounding-aware outlier channel splitting, which mitigates quantization error while acting as an identity transform that preserves the quantized outputs. Furthermore, we follow microscaling groups with E4M3 scales, capturing dynamic activation ranges in alignment with OCP/NVIDIA standards. To address the lack of efficient 2-bit kernels, we developed custom operators for both W2A2 and W2A16 configurations, achieving up to 11times speedup over BF16. Under W2A2 settings, Bit-by-Bit significantly outperforms baselines like BitDistiller and EfficientQAT on both Llama2/3, achieving a loss of only 2.25 WikiText2 PPL compared to full-precision models.

Financial time series forecasting is particularly challenging for transformer-based time series foundation models (TSFMs) due to non-stationarity, heavy-tailed distributions, and high-frequency noise present in data. Low-rank adaptation (LoRA) has become a popular parameter-efficient method for adapting pre-trained TSFMs to downstream data domains. However, it still underperforms in financial data, as it preserves the network architecture and training objective of TSFMs rather than complementing the foundation model. To further enhance TSFMs, we propose a novel refinement module, RefineBridge, built upon a tractable Schrödinger Bridge (SB) generative framework. Given the forecasts of TSFM as generative prior and the observed ground truths as targets, RefineBridge learns context-conditioned stochastic transport maps to improve TSFM predictions, iteratively approaching the ground-truth target from even a low-quality prior. Simulations on multiple financial benchmarks demonstrate that RefineBridge consistently improves the performance of state-of-the-art TSFMs across different prediction horizons.

Dimensionality reduction is critical for deploying dense retrieval systems at scale, yet mainstream post-hoc methods face a fundamental trade-off: principal component analysis (PCA) preserves dominant variance but underutilizes representational capacity, while whitening enforces isotropy at the cost of amplifying noise in the heavy-tailed eigenspectrum of retrieval embeddings. Intermediate spectral scaling methods unify these extremes by reweighting dimensions with a power coefficient γ, but treat γ as a fixed hyperparameter that requires task-specific tuning. We show that the optimal scaling strength γ is not a global constant: it varies systematically with target dimensionality k and is governed by the signal-to-noise ratio (SNR) of the retained subspace. Based on this insight, we propose Spectral Tempering (SpecTemp), a learning-free method that derives an adaptive γ(k) directly from the corpus eigenspectrum using local SNR analysis and knee-point normalization, requiring no labeled data or validation-based search. Extensive experiments demonstrate that Spectral Tempering consistently achieves near-oracle performance relative to grid-searched γ^*(k) while remaining fully learning-free and model-agnostic. Our code is publicly available at https://anonymous.4open.science/r/SpecTemp-0D37.

High-Dimensional Low-Sample Size (HDLSS) tabular domains (e.g., omics) are characterized by n ll m, where n = number of samples, and m = number of features. Such domains often exhibit strong local correlation groups, sparse cross-group dependencies, heavy-tailed non-Gaussian marginals, heteroscedastic noise, and structured missingness, making direct density learning in R^m ill-conditioned since n ll m. We propose BSTabDiff, a block-subunit generative framework that partitions the m observed features into M latent blocks (M ll m) and generates each block via a shared low-dimensional subunit variable, concentrating global dependence learning in the compact block-latent space R^M while decoding to the full feature space with copula-driven dependence, flexible per-feature marginals, and explicit missingness mechanisms. BSTabDiff supports modern deep priors on block latents, including diffusion and normalizing flows, enabling stable synthesis and controllable benchmark generation in the HDLSS regime. Empirically, BSTabDiff produces more realistic and stable high-dimensional synthetic data when compared with unstructured tabular generators on HDLSS data.

The complicated architecture and high training cost of vision transformers urge the exploration of post-training quantization. However, the heavy-tailed distribution of vision transformer activations hinders the effectiveness of previous post-training quantization methods, even with advanced quantizer designs. Instead of tuning the quantizer to better fit the complicated activation distribution, this paper proposes NoisyQuant, a quantizer-agnostic enhancement for the post-training activation quantization performance of vision transformers. We make a surprising theoretical discovery that for a given quantizer, adding a fixed Uniform noisy bias to the values being quantized can significantly reduce the quantization error under provable conditions. Building on the theoretical insight, NoisyQuant achieves the first success on actively altering the heavy-tailed activation distribution with additive noisy bias to fit a given quantizer. Extensive experiments show NoisyQuant largely improves the post-training quantization performance of vision transformer with minimal computation overhead. For instance, on linear uniform 6-bit activation quantization, NoisyQuant improves SOTA top-1 accuracy on ImageNet by up to 1.7%, 1.1% and 0.5% for ViT, DeiT, and Swin Transformer respectively, achieving on-par or even higher performance than previous nonlinear, mixed-precision quantization.

Recent works have revealed that infinitely-wide feed-forward or recurrent neural networks of any architecture correspond to Gaussian processes referred to as Neural Network Gaussian Processes (NNGPs). While these works have extended the class of neural networks converging to Gaussian processes significantly, however, there has been little focus on broadening the class of stochastic processes that such neural networks converge to. In this work, inspired by the scale mixture of Gaussian random variables, we propose the scale mixture of NNGPs for which we introduce a prior distribution on the scale of the last-layer parameters. We show that simply introducing a scale prior on the last-layer parameters can turn infinitely-wide neural networks of any architecture into a richer class of stochastic processes. With certain scale priors, we obtain heavy-tailed stochastic processes, and in the case of inverse gamma priors, we recover Student's t processes. We further analyze the distributions of the neural networks initialized with our prior setting and trained with gradient descents and obtain similar results as for NNGPs. We present a practical posterior-inference algorithm for the scale mixture of NNGPs and empirically demonstrate its usefulness on regression and classification tasks. In particular, we show that in both tasks, the heavy-tailed stochastic processes obtained from our framework are robust to out-of-distribution data.

Heavy-tail phenomena in stochastic gradient descent (SGD) have been reported in several empirical studies. Experimental evidence in previous works suggests a strong interplay between the heaviness of the tails and generalization behavior of SGD. To address this empirical phenomena theoretically, several works have made strong topological and statistical assumptions to link the generalization error to heavy tails. Very recently, new generalization bounds have been proven, indicating a non-monotonic relationship between the generalization error and heavy tails, which is more pertinent to the reported empirical observations. While these bounds do not require additional topological assumptions given that SGD can be modeled using a heavy-tailed stochastic differential equation (SDE), they can only apply to simple quadratic problems. In this paper, we build on this line of research and develop generalization bounds for a more general class of objective functions, which includes non-convex functions as well. Our approach is based on developing Wasserstein stability bounds for heavy-tailed SDEs and their discretizations, which we then convert to generalization bounds. Our results do not require any nontrivial assumptions; yet, they shed more light to the empirical observations, thanks to the generality of the loss functions.

This paper studies Kernel density estimation for a high-dimensional distribution rho(x). Traditional approaches have focused on the limit of large number of data points n and fixed dimension d. We analyze instead the regime where both the number n of data points y_i and their dimensionality d grow with a fixed ratio alpha=(log n)/d. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density hat rho_h^{D}(x)=1{n h^d}sum_{i=1}^n Kleft(x-y_i{h}right), depending on the bandwidth h: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, h_{CLT}(alpha), we find that the CLT breaks down. The statistics of hat rho_h^{D}(x) for a fixed x drawn from rho(x) is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value h_G(alpha), we find that hat rho_h^{D}(x) is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. Our findings reveal limitations of classical approaches, show the relevance of these new statistical regimes, and offer new insights for Kernel density estimation in high-dimensional settings.

Multiplicative noise, also known as speckle or pepper noise, commonly affects images produced by synthetic aperture radar (SAR), lasers, or optical lenses. Unlike additive noise, which typically arises from thermal processes or external factors, multiplicative noise is inherent to the system, originating from the fluctuation in diffuse reflections. These fluctuations result in multiple copies of the same signal with varying magnitudes being combined. Consequently, despeckling, or removing multiplicative noise, necessitates different techniques compared to those used for additive noise removal. In this paper, we propose a novel approach using Stochastic Differential Equations based diffusion models to address multiplicative noise. We demonstrate that multiplicative noise can be effectively modeled as a Geometric Brownian Motion process in the logarithmic domain. Utilizing the Fokker-Planck equation, we derive the corresponding reverse process for image denoising. To validate our method, we conduct extensive experiments on two different datasets, comparing our approach to both classical signal processing techniques and contemporary CNN-based noise removal models. Our results indicate that the proposed method significantly outperforms existing methods on perception-based metrics such as FID and LPIPS, while maintaining competitive performance on traditional metrics like PSNR and SSIM.

We develop an econometric framework integrating heavy-tailed Student's t distributions with behavioral probability weighting while preserving infinite divisibility. Using 432{,}752 observations across 86 assets (2004--2024), we demonstrate Student's t specifications outperform Gaussian models in 88.4\% of cases. Bounded probability-weighting transformations preserve mathematical properties required for dynamic pricing. Gaussian models underestimate 99\% Value-at-Risk by 19.7\% versus 3.2\% for our specification. Joint estimation procedures identify tail and behavioral parameters with established asymptotic properties. Results provide robust inference for asset-pricing applications where heavy tails and behavioral distortions coexist.

Long-tailed classification is challenging due to its heavy imbalance in class probabilities. While existing methods often focus on overall accuracy or accuracy for tail classes, they overlook a critical aspect: certain types of errors can carry greater risks than others in real-world long-tailed problems. For example, misclassifying patients (a tail class) as healthy individuals (a head class) entails far more serious consequences than the reverse scenario. To address this critical issue, we introduce Making Reliable and Flexible Decisions in Long-tailed Classification (RF-DLC), a novel framework aimed at reliable predictions in long-tailed problems. Leveraging Bayesian Decision Theory, we introduce an integrated gain to seamlessly combine long-tailed data distributions and the decision-making procedure. We further propose an efficient variational optimization strategy for the decision risk objective. Our method adapts readily to diverse utility matrices, which can be designed for specific tasks, ensuring its flexibility for different problem settings. In empirical evaluation, we design a new metric, False Head Rate, to quantify tail-sensitivity risk, along with comprehensive experiments on multiple real-world tasks, including large-scale image classification and uncertainty quantification, to demonstrate the reliability and flexibility of our method.

A collection of robust Mahalanobis distances for multivariate outlier detection is proposed, based on the notion of shrinkage. Robust intensity and scaling factors are optimally estimated to define the shrinkage. Some properties are investigated, such as affine equivariance and breakdown value. The performance of the proposal is illustrated through the comparison to other techniques from the literature, in a simulation study and with a real dataset. The behavior when the underlying distribution is heavy-tailed or skewed, shows the appropriateness of the method when we deviate from the common assumption of normality. The resulting high correct detection rates and low false detection rates in the vast majority of cases, as well as the significantly smaller computation time shows the advantages of our proposal.

Learning to denoise has emerged as a prominent paradigm to design state-of-the-art deep generative models for natural images. How to use it to model the distributions of both continuous real-valued data and categorical data has been well studied in recently proposed diffusion models. However, it is found in this paper to have limited ability in modeling some other types of data, such as count and non-negative continuous data, that are often highly sparse, skewed, heavy-tailed, and/or overdispersed. To this end, we propose learning to jump as a general recipe for generative modeling of various types of data. Using a forward count thinning process to construct learning objectives to train a deep neural network, it employs a reverse count thickening process to iteratively refine its generation through that network. We demonstrate when learning to jump is expected to perform comparably to learning to denoise, and when it is expected to perform better. For example, learning to jump is recommended when the training data is non-negative and exhibits strong sparsity, skewness, heavy-tailedness, and/or heterogeneity.

We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect Y can be written as a function of the cause X and a noise source N independent of X, which may be scaled by a positive function g over the cause, i.e., Y = f(X) + g(X)N. Despite the generality of the model class, we show the causal direction is identifiable up to some pathological cases. To empirically validate these theoretical findings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on neural networks. Both model the conditional distribution of Y given X as a Gaussian parameterized by its natural parameters. When the feature maps are correctly specified, we prove that our estimator is jointly concave, and a consistent estimator for the cause-effect identification task. Although the the neural network does not inherit those guarantees, it can fit functions of arbitrary complexity, and reaches state-of-the-art performance across benchmarks.

Long-tailed classification poses a challenge due to its heavy imbalance in class probabilities and tail-sensitivity risks with asymmetric misprediction costs. Recent attempts have used re-balancing loss and ensemble methods, but they are largely heuristic and depend heavily on empirical results, lacking theoretical explanation. Furthermore, existing methods overlook the decision loss, which characterizes different costs associated with tailed classes. This paper presents a general and principled framework from a Bayesian-decision-theory perspective, which unifies existing techniques including re-balancing and ensemble methods, and provides theoretical justifications for their effectiveness. From this perspective, we derive a novel objective based on the integrated risk and a Bayesian deep-ensemble approach to improve the accuracy of all classes, especially the "tail". Besides, our framework allows for task-adaptive decision loss which provides provably optimal decisions in varying task scenarios, along with the capability to quantify uncertainty. Finally, We conduct comprehensive experiments, including standard classification, tail-sensitive classification with a new False Head Rate metric, calibration, and ablation studies. Our framework significantly improves the current SOTA even on large-scale real-world datasets like ImageNet.

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the part of the distribution representing large but rare events -- and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out.

Learning rate configuration is a fundamental aspect of modern deep learning. The prevailing practice of applying a uniform learning rate across all layers overlooks the structural heterogeneity of Transformers, potentially limiting their effectiveness as the backbone of Large Language Models (LLMs). In this paper, we introduce Layerwise Learning Rate (LLR), an adaptive scheme that assigns distinct learning rates to individual Transformer layers. Our method is grounded in Heavy-Tailed Self-Regularization (HT-SR) theory, which characterizes the empirical spectral density (ESD) of weight correlation matrices to quantify heavy-tailedness. Layers with weaker heavy-tailedness are assigned larger learning rates to accelerate training, while layers with stronger heavy-tailedness receive smaller learning rates. By tailoring learning rates in this manner, LLR promotes more balanced training across layers, leading to faster convergence and improved generalization. Extensive experiments across architectures ranging from LLaMA to GPT-nano, optimizers including AdamW and Muon, and model scales from 60M to 3B parameters with up to 100B training tokens demonstrate the effectiveness of LLR. LLR achieves up to 1.5x training speedup and consistently outperforms uniform-learning-rate baselines. In particular, it improves the average zero-shot accuracy of 1B models from 47.09% to 49.02%, and that of 3B models from 48.58% to 50.61%. A key advantage of LLR is its low tuning overhead: it can transfer nearly optimal learning-rate settings directly from the uniform baseline. Code is available at https://github.com/hed-ucas/Layer-wise-Learning-Rate.

When facing data with imbalanced classes or groups, practitioners follow an intriguing strategy to achieve best results. They throw away examples until the classes or groups are balanced in size, and then perform empirical risk minimization on the reduced training set. This opposes common wisdom in learning theory, where the expected error is supposed to decrease as the dataset grows in size. In this work, we leverage extreme value theory to address this apparent contradiction. Our results show that the tails of the data distribution play an important role in determining the worst-group-accuracy of linear classifiers. When learning on data with heavy tails, throwing away data restores the geometric symmetry of the resulting classifier, and therefore improves its worst-group generalization.

This paper investigates the problem of generalized linear bandits with heavy-tailed rewards, whose (1+epsilon)-th moment is bounded for some epsilonin (0,1]. Although there exist methods for generalized linear bandits, most of them focus on bounded or sub-Gaussian rewards and are not well-suited for many real-world scenarios, such as financial markets and web-advertising. To address this issue, we propose two novel algorithms based on truncation and mean of medians. These algorithms achieve an almost optimal regret bound of O(dT^{1{1+epsilon}}), where d is the dimension of contextual information and T is the time horizon. Our truncation-based algorithm supports online learning, distinguishing it from existing truncation-based approaches. Additionally, our mean-of-medians-based algorithm requires only O(log T) rewards and one estimator per epoch, making it more practical. Moreover, our algorithms improve the regret bounds by a logarithmic factor compared to existing algorithms when epsilon=1. Numerical experimental results confirm the merits of our algorithms.

Learning the tail behavior of a distribution is a notoriously difficult problem. By definition, the number of samples from the tail is small, and deep generative models, such as normalizing flows, tend to concentrate on learning the body of the distribution. In this paper, we focus on improving the ability of normalizing flows to correctly capture the tail behavior and, thus, form more accurate models. We prove that the marginal tailedness of an autoregressive flow can be controlled via the tailedness of the marginals of its base distribution. This theoretical insight leads us to a novel type of flows based on flexible base distributions and data-driven linear layers. An empirical analysis shows that the proposed method improves on the accuracy -- especially on the tails of the distribution -- and is able to generate heavy-tailed data. We demonstrate its application on a weather and climate example, in which capturing the tail behavior is essential.

Real-world data tends to follow a long-tailed distribution, where the class imbalance results in dominance of the head classes during training. In this paper, we propose a frustratingly simple but effective step-wise learning framework to gradually enhance the capability of the model in detecting all categories of long-tailed datasets. Specifically, we build smooth-tail data where the long-tailed distribution of categories decays smoothly to correct the bias towards head classes. We pre-train a model on the whole long-tailed data to preserve discriminability between all categories. We then fine-tune the class-agnostic modules of the pre-trained model on the head class dominant replay data to get a head class expert model with improved decision boundaries from all categories. Finally, we train a unified model on the tail class dominant replay data while transferring knowledge from the head class expert model to ensure accurate detection of all categories. Extensive experiments on long-tailed datasets LVIS v0.5 and LVIS v1.0 demonstrate the superior performance of our method, where we can improve the AP with ResNet-50 backbone from 27.0% to 30.3% AP, and especially for the rare categories from 15.5% to 24.9% AP. Our best model using ResNet-101 backbone can achieve 30.7% AP, which suppresses all existing detectors using the same backbone.

Marcinkiewicz strong law of large numbers, {n^{-frac1p}}sum_{k=1}^{n} (d_{k}- d)rightarrow 0 almost surely with pin(1,2), are developed for products d_k=prod_{r=1}^s x_k^{(r)}, where the x_k^{(r)} = sum_{l=-infty}^{infty}c_{k-l}^{(r)}xi_l^{(r)} are two-sided linear processes with coefficients {c_l^{(r)}}_{lin Z} and i.i.d. zero-mean innovations {xi_l^{(r)}}_{lin Z}. The decay of the coefficients c_l^{(r)} as |l|toinfty, can be slow enough for {x_k^{(r)}} to have long memory while {d_k} can have heavy tails. The long-range dependence and heavy tails for {d_k} are handled simultaneously and a decoupling property shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The Marcinkiewicz strong law of large numbers is also extended to the multivariate linear process case.

We consider the problem of computing the Walsh-Hadamard Transform (WHT) of some N-length input vector in the presence of noise, where the N-point Walsh spectrum is K-sparse with K = {O}(N^{delta}) scaling sub-linearly in the input dimension N for some 0<delta<1. Over the past decade, there has been a resurgence in research related to the computation of Discrete Fourier Transform (DFT) for some length-N input signal that has a K-sparse Fourier spectrum. In particular, through a sparse-graph code design, our earlier work on the Fast Fourier Aliasing-based Sparse Transform (FFAST) algorithm computes the K-sparse DFT in time {O}(Klog K) by taking {O}(K) noiseless samples. Inspired by the coding-theoretic design framework, Scheibler et al. proposed the Sparse Fast Hadamard Transform (SparseFHT) algorithm that elegantly computes the K-sparse WHT in the absence of noise using {O}(Klog N) samples in time {O}(Klog^2 N). However, the SparseFHT algorithm explicitly exploits the noiseless nature of the problem, and is not equipped to deal with scenarios where the observations are corrupted by noise. Therefore, a question of critical interest is whether this coding-theoretic framework can be made robust to noise. Further, if the answer is yes, what is the extra price that needs to be paid for being robust to noise? In this paper, we show, quite interestingly, that there is {\it no extra price} that needs to be paid for being robust to noise other than a constant factor. In other words, we can maintain the same sample complexity {O}(Klog N) and the computational complexity {O}(Klog^2 N) as those of the noiseless case, using our SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm.

The long-tailed distribution is a common phenomenon in the real world. Extracted large scale image datasets inevitably demonstrate the long-tailed property and models trained with imbalanced data can obtain high performance for the over-represented categories, but struggle for the under-represented categories, leading to biased predictions and performance degradation. To address this challenge, we propose a novel de-biasing method named Inverse Image Frequency (IIF). IIF is a multiplicative margin adjustment transformation of the logits in the classification layer of a convolutional neural network. Our method achieves stronger performance than similar works and it is especially useful for downstream tasks such as long-tailed instance segmentation as it produces fewer false positive detections. Our extensive experiments show that IIF surpasses the state of the art on many long-tailed benchmarks such as ImageNet-LT, CIFAR-LT, Places-LT and LVIS, reaching 55.8% top-1 accuracy with ResNet50 on ImageNet-LT and 26.2% segmentation AP with MaskRCNN on LVIS. Code available at https://github.com/kostas1515/iif

Internet photo collections exhibit an extremely long-tailed distribution: a few famous landmarks are densely photographed and easily reconstructed in 3D, while most real-world sites are represented with sparse, noisy, uneven imagery beyond the capabilities of both classical and learned 3D methods. We believe that tackling this long-tail regime represents one of the next frontiers for 3D foundation models. Although reliable ground-truth 3D supervision from sparse scenes is challenging to acquire, we observe that it can be effectively simulated by sampling sparse subsets from well-reconstructed Internet landmarks. To this end, we introduce MegaDepth-X, a large dataset of 3D reconstructions with clean, dense depth, together with a strategy for sampling sets of training images that mimic camera distributions in long-tail scenes. Finetuning 3D foundation models with these components yields robust reconstructions under extreme sparsity, and also enables more reliable reconstruction in symmetric and repetitive scenes, while preserving generalization to standard, dense 3D benchmark datasets.

We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for sigma-sub-Gaussian label noise, our analysis provides a regret upper bound of O(sigma^2 d log T) + o(log T), where d is the dimension of the input vector, T is the total number of rounds. We also prove a Omega(sigma^2dlog(T/d)) lower bound for stochastic online linear regression, which indicates that our upper bound is nearly optimal. In addition, we extend our analysis to a more refined Bernstein noise condition. As an application, we study generalized linear bandits with heteroscedastic noise and propose an algorithm based on FTRL to achieve the first variance-aware regret bound.

Modeling and synthesizing low-light raw noise is a fundamental problem for computational photography and image processing applications. Although most recent works have adopted physics-based models to synthesize noise, the signal-independent noise in low-light conditions is far more complicated and varies dramatically across camera sensors, which is beyond the description of these models. To address this issue, we introduce a new perspective to synthesize the signal-independent noise by a generative model. Specifically, we synthesize the signal-dependent and signal-independent noise in a physics- and learning-based manner, respectively. In this way, our method can be considered as a general model, that is, it can simultaneously learn different noise characteristics for different ISO levels and generalize to various sensors. Subsequently, we present an effective multi-scale discriminator termed Fourier transformer discriminator (FTD) to distinguish the noise distribution accurately. Additionally, we collect a new low-light raw denoising (LRD) dataset for training and benchmarking. Qualitative validation shows that the noise generated by our proposed noise model can be highly similar to the real noise in terms of distribution. Furthermore, extensive denoising experiments demonstrate that our method performs favorably against state-of-the-art methods on different sensors.

Classification on long-tailed distributed data is a challenging problem, which suffers from serious class-imbalance and accordingly unpromising performance especially on tail classes. Recently, the ensembling based methods achieve the state-of-the-art performance and show great potential. However, there are two limitations for current methods. First, their predictions are not trustworthy for failure-sensitive applications. This is especially harmful for the tail classes where the wrong predictions is basically frequent. Second, they assign unified numbers of experts to all samples, which is redundant for easy samples with excessive computational cost. To address these issues, we propose a Trustworthy Long-tailed Classification (TLC) method to jointly conduct classification and uncertainty estimation to identify hard samples in a multi-expert framework. Our TLC obtains the evidence-based uncertainty (EvU) and evidence for each expert, and then combines these uncertainties and evidences under the Dempster-Shafer Evidence Theory (DST). Moreover, we propose a dynamic expert engagement to reduce the number of engaged experts for easy samples and achieve efficiency while maintaining promising performances. Finally, we conduct comprehensive experiments on the tasks of classification, tail detection, OOD detection and failure prediction. The experimental results show that the proposed TLC outperforms existing methods and is trustworthy with reliable uncertainty.

We show via a combination of mathematical arguments and empirical evidence that data distributions sampled from diffusion models satisfy a Concentration of Measure Property saying that any Lipschitz 1-dimensional projection of a random vector is not too far from its mean with high probability. This implies that such models are quite restrictive and gives an explanation for a fact previously observed in the literature that conventional diffusion models cannot capture "heavy-tailed" data (i.e. data x for which the norm |x|_2 does not possess a sub-Gaussian tail) well. We then proceed to train a generalized linear model using stochastic gradient descent (SGD) on the diffusion-generated data for a multiclass classification task and observe empirically that a Gaussian universality result holds for the test error. In other words, the test error depends only on the first and second order statistics of the diffusion-generated data in the linear setting. Results of such forms are desirable because they allow one to assume the data itself is Gaussian for analyzing performance of the trained classifier. Finally, we note that current approaches to proving universality do not apply to this case as the covariance matrices of the data tend to have vanishing minimum singular values for the diffusion-generated data, while the current proofs assume that this is not the case (see Subsection 3.4 for more details). This leaves extending previous mathematical universality results as an intriguing open question.

While implicit regularization facilitates benign overfitting in low-noise regimes, recent theoretical work predicts a sharp phase transition to harmful overfitting as the noise-to-signal ratio increases. We experimentally isolate the geometric mechanism of this transition: the Malignant Tail, a failure mode where networks functionally segregate signal and noise, reducing coherent semantic features into low-rank subspaces while pushing stochastic label noise into high-frequency orthogonal components, distinct from systematic or corruption-aligned noise. Through a Spectral Linear Probe of training dynamics, we demonstrate that Stochastic Gradient Descent (SGD) fails to suppress this noise, instead implicitly biasing it toward high-frequency orthogonal subspaces, effectively preserving signal-noise separability. We show that this geometric separation is distinct from simple variance reduction in untrained models. In trained networks, SGD actively segregates noise, allowing post-hoc Explicit Spectral Truncation (d << D) to surgically prune the noise-dominated subspace. This approach recovers the optimal generalization capability latent in the converged model. Unlike unstable temporal early stopping, Geometric Truncation provides a stable post-hoc intervention. Our findings suggest that under label noise, excess spectral capacity is not harmless redundancy but a latent structural liability that allows for noise memorization, necessitating explicit rank constraints to filter stochastic corruptions for robust generalization.

We study generative modeling on convex domains using flow matching and mirror maps, and identify two fundamental challenges. First, standard log-barrier mirror maps induce heavy-tailed dual distributions, leading to ill-posed dynamics. Second, coupling with Gaussian priors performs poorly when matching heavy-tailed targets. To address these issues, we propose Mirror Flow Matching based on a regularized mirror map that controls dual tail behavior and guarantees finite moments, together with coupling to a Student-t prior that aligns with heavy-tailed targets and stabilizes training. We provide theoretical guarantees, including spatial Lipschitzness and temporal regularity of the velocity field, Wasserstein convergence rates for flow matching with Student-t priors and primal-space guarantees for constrained generation, under varepsilon-accurate learned velocity fields. Empirically, our method outperforms baselines in synthetic convex-domain simulations and achieves competitive sample quality on real-world constrained generative tasks.

Understanding when the noise in stochastic gradient descent (SGD) affects generalization of deep neural networks remains a challenge, complicated by the fact that networks can operate in distinct training regimes. Here we study how the magnitude of this noise T affects performance as the size of the training set P and the scale of initialization alpha are varied. For gradient descent, alpha is a key parameter that controls if the network is `lazy'(alphagg1) or instead learns features (alphall1). For classification of MNIST and CIFAR10 images, our central results are: (i) obtaining phase diagrams for performance in the (alpha,T) plane. They show that SGD noise can be detrimental or instead useful depending on the training regime. Moreover, although increasing T or decreasing alpha both allow the net to escape the lazy regime, these changes can have opposite effects on performance. (ii) Most importantly, we find that the characteristic temperature T_c where the noise of SGD starts affecting the trained model (and eventually performance) is a power law of P. We relate this finding with the observation that key dynamical quantities, such as the total variation of weights during training, depend on both T and P as power laws. These results indicate that a key effect of SGD noise occurs late in training by affecting the stopping process whereby all data are fitted. Indeed, we argue that due to SGD noise, nets must develop a stronger `signal', i.e. larger informative weights, to fit the data, leading to a longer training time. A stronger signal and a longer training time are also required when the size of the training set P increases. We confirm these views in the perceptron model, where signal and noise can be precisely measured. Interestingly, exponents characterizing the effect of SGD depend on the density of data near the decision boundary, as we explain.

Text-to-image diffusion model is a popular paradigm that synthesizes personalized images by providing a text prompt and a random Gaussian noise. While people observe that some noises are ``golden noises'' that can achieve better text-image alignment and higher human preference than others, we still lack a machine learning framework to obtain those golden noises. To learn golden noises for diffusion sampling, we mainly make three contributions in this paper. First, we identify a new concept termed the noise prompt, which aims at turning a random Gaussian noise into a golden noise by adding a small desirable perturbation derived from the text prompt. Following the concept, we first formulate the noise prompt learning framework that systematically learns ``prompted'' golden noise associated with a text prompt for diffusion models. Second, we design a noise prompt data collection pipeline and collect a large-scale noise prompt dataset~(NPD) that contains 100k pairs of random noises and golden noises with the associated text prompts. With the prepared NPD as the training dataset, we trained a small noise prompt network~(NPNet) that can directly learn to transform a random noise into a golden noise. The learned golden noise perturbation can be considered as a kind of prompt for noise, as it is rich in semantic information and tailored to the given text prompt. Third, our extensive experiments demonstrate the impressive effectiveness and generalization of NPNet on improving the quality of synthesized images across various diffusion models, including SDXL, DreamShaper-xl-v2-turbo, and Hunyuan-DiT. Moreover, NPNet is a small and efficient controller that acts as a plug-and-play module with very limited additional inference and computational costs, as it just provides a golden noise instead of a random noise without accessing the original pipeline.

During both pretraining and fine-tuning, Large Language Models (LLMs) are trained on trillions of tokens of text of widely varying quality. Both phases of training typically involve heuristically filtering out ``low-quality'' or noisy training samples, yet little is known quantitatively about how the type or intensity of noise affects downstream performance. In this work, we study how noise in chain of thought (CoT) impacts task performance in the highly-controlled setting of algorithmically solvable tasks. First, we develop the Traced Integer (TInt) framework to generate highly customizable noised execution traces for any arithmetic function on lists of integers. We then define two types of noise: static noise, a local form of noise which is applied after the CoT trace is computed, and dynamic noise, a global form of noise which propagates errors in the trace as it is computed. We then evaluate the test performance of pretrained models both prompted and fine-tuned on noised datasets with varying levels of dataset contamination and intensity. We find fine-tuned models are extremely robust to high levels of static noise but struggle significantly more with lower levels of dynamic noise. In contrast, few-shot prompted models appear more sensitive to even static noise. We conclude with a discussion of how our findings impact noise filtering best-practices, in particular emphasizing the importance of removing samples containing destructive dynamic noise with global errors.

We introduce a family of random matrices where correlations between matrix elements are induced via interaction-derived Boltzmann factors. Varying these yields access to different ensembles. We find a universal scaling behavior of the finite-size statistics characterized by a heavy-tailed eigenvalue distribution whose extremes are governed by the Fréchet extreme value distribution for the case corresponding to a ferromagnetic Ising transition. The introduction of a finite density of nonlocal interactions restores standard random-matrix behavior. Suitably rescaled average extremes, playing a physical role as an order parameter, can thus discriminate aspects of the interaction structure; they also yield further nonuniversal information. In particular, the link between maximum eigenvalues and order parameters offers a potential route to resolving long-standing problems in statistical physics, such as deriving the exact magnetization scaling function in the two-dimensional Ising model at criticality.

Decentralized learning enables distributed agents to collaboratively train a shared machine learning model without a central server, through local computation and peer-to-peer communication. Although each agent retains its dataset locally, sharing local models can still expose private information about the local training datasets to adversaries. To mitigate privacy attacks, a common strategy is to inject random artificial noise at each agent before exchanging local models between neighbors. However, this often leads to utility degradation due to the negative effects of cumulated artificial noise on the learning algorithm. In this work, we introduce CorN-DSGD, a novel covariance-based framework for generating correlated privacy noise across agents, which unifies several state-of-the-art methods as special cases. By leveraging network topology and mixing weights, CorN-DSGD optimizes the noise covariance to achieve network-wide noise cancellation. Experimental results show that CorN-DSGD cancels more noise than existing pairwise correlation schemes, improving model performance under formal privacy guarantees.

Enhancing the visibility in extreme low-light environments is a challenging task. Under nearly lightless condition, existing image denoising methods could easily break down due to significantly low SNR. In this paper, we systematically study the noise statistics in the imaging pipeline of CMOS photosensors, and formulate a comprehensive noise model that can accurately characterize the real noise structures. Our novel model considers the noise sources caused by digital camera electronics which are largely overlooked by existing methods yet have significant influence on raw measurement in the dark. It provides a way to decouple the intricate noise structure into different statistical distributions with physical interpretations. Moreover, our noise model can be used to synthesize realistic training data for learning-based low-light denoising algorithms. In this regard, although promising results have been shown recently with deep convolutional neural networks, the success heavily depends on abundant noisy clean image pairs for training, which are tremendously difficult to obtain in practice. Generalizing their trained models to images from new devices is also problematic. Extensive experiments on multiple low-light denoising datasets -- including a newly collected one in this work covering various devices -- show that a deep neural network trained with our proposed noise formation model can reach surprisingly-high accuracy. The results are on par with or sometimes even outperform training with paired real data, opening a new door to real-world extreme low-light photography.

We examine how various types of noise in the parallel training data impact the quality of neural machine translation systems. We create five types of artificial noise and analyze how they degrade performance in neural and statistical machine translation. We find that neural models are generally more harmed by noise than statistical models. For one especially egregious type of noise they learn to just copy the input sentence.

In this work, we consider rather general and broad class of Markov chains, Ito chains, that look like Euler-Maryama discretization of some Stochastic Differential Equation. The chain we study is a unified framework for theoretical analysis. It comes with almost arbitrary isotropic and state-dependent noise instead of normal and state-independent one as in most related papers. Moreover, in our chain the drift and diffusion coefficient can be inexact in order to cover wide range of applications as Stochastic Gradient Langevin Dynamics, sampling, Stochastic Gradient Descent or Stochastic Gradient Boosting. We prove the bound in W_{2}-distance between the laws of our Ito chain and corresponding differential equation. These results improve or cover most of the known estimates. And for some particular cases, our analysis is the first.

Long-tailed image classification remains a long-standing challenge, as real-world data typically follow highly imbalanced distributions where a few head classes dominate and many tail classes contain only limited samples. This imbalance biases feature learning toward head categories and leads to significant degradation on rare classes. Although recent studies have proposed re-sampling, re-weighting, and decoupled learning strategies, the improvement on the most underrepresented classes still remains marginal compared with overall accuracy. In this work, we present a confusion-centric perspective for long-tailed recognition that explicitly focuses on worst-class generalization. We first establish a new theoretical framework of class-specific error analysis, which shows that the worst-class error can be tightly upper-bounded by the spectral norm of the frequency-weighted confusion matrix and a model-dependent complexity term. Guided by this insight, we propose the Confusion-Aware Spectral Regularizer (CAR) that minimizes the spectral norm of the confusion matrix during training to reduce inter-class confusion and enhance tail-class generalization. To enable stable and efficient optimization, CAR integrates a Differentiable Confusion Matrix Surrogate and an EMA-based Confusion Estimator to maintain smooth and low-variance estimates across mini-batches. Extensive experiments across multiple long-tailed benchmarks demonstrates that CAR substantially improves both worst-class accuracy and overall performance. When combined with ConCutMix augmentation, CAR consistently surpasses exisiting state-of-the-art long-tailed learning methods under both the training-from-scratch setting (by 2.37% ~ 4.83%) and the fine-tuning-from-pretrained setting (by 2.42% ~ 4.17%) across ImageNet-LT, CIFAR100-LT, and iNaturalist datasets.

Diffusion models have opened the path to a wide range of text-based image editing frameworks. However, these typically build on the multi-step nature of the diffusion backwards process, and adapting them to distilled, fast-sampling methods has proven surprisingly challenging. Here, we focus on a popular line of text-based editing frameworks - the ``edit-friendly'' DDPM-noise inversion approach. We analyze its application to fast sampling methods and categorize its failures into two classes: the appearance of visual artifacts, and insufficient editing strength. We trace the artifacts to mismatched noise statistics between inverted noises and the expected noise schedule, and suggest a shifted noise schedule which corrects for this offset. To increase editing strength, we propose a pseudo-guidance approach that efficiently increases the magnitude of edits without introducing new artifacts. All in all, our method enables text-based image editing with as few as three diffusion steps, while providing novel insights into the mechanisms behind popular text-based editing approaches.

Recently, there has been extensive research interest in training deep networks to denoise images without clean reference. However, the representative approaches such as Noise2Noise, Noise2Void, Stein's unbiased risk estimator (SURE), etc. seem to differ from one another and it is difficult to find the coherent mathematical structure. To address this, here we present a novel approach, called Noise2Score, which reveals a missing link in order to unite these seemingly different approaches. Specifically, we show that image denoising problems without clean images can be addressed by finding the mode of the posterior distribution and that the Tweedie's formula offers an explicit solution through the score function (i.e. the gradient of log likelihood). Our method then uses the recent finding that the score function can be stably estimated from the noisy images using the amortized residual denoising autoencoder, the method of which is closely related to Noise2Noise or Nose2Void. Our Noise2Score approach is so universal that the same network training can be used to remove noises from images that are corrupted by any exponential family distributions and noise parameters. Using extensive experiments with Gaussian, Poisson, and Gamma noises, we show that Noise2Score significantly outperforms the state-of-the-art self-supervised denoising methods in the benchmark data set such as (C)BSD68, Set12, and Kodak, etc.

Tweedie distributions are a special case of exponential dispersion models, which are often used in classical statistics as distributions for generalized linear models. Here, we reveal that Tweedie distributions also play key roles in modern deep learning era, leading to a distribution independent self-supervised image denoising formula without clean reference images. Specifically, by combining with the recent Noise2Score self-supervised image denoising approach and the saddle point approximation of Tweedie distribution, we can provide a general closed-form denoising formula that can be used for large classes of noise distributions without ever knowing the underlying noise distribution. Similar to the original Noise2Score, the new approach is composed of two successive steps: score matching using perturbed noisy images, followed by a closed form image denoising formula via distribution-independent Tweedie's formula. This also suggests a systematic algorithm to estimate the noise model and noise parameters for a given noisy image data set. Through extensive experiments, we demonstrate that the proposed method can accurately estimate noise models and parameters, and provide the state-of-the-art self-supervised image denoising performance in the benchmark dataset and real-world dataset.

Label noise is one of the key factors that lead to the poor generalization of deep learning models. Existing label-noise learning methods usually assume that the ground-truth classes of the training data are balanced. However, the real-world data is often imbalanced, leading to the inconsistency between observed and intrinsic class distribution with label noises. In this case, it is hard to distinguish clean samples from noisy samples on the intrinsic tail classes with the unknown intrinsic class distribution. In this paper, we propose a learning framework for label-noise learning with intrinsically long-tailed data. Specifically, we propose two-stage bi-dimensional sample selection (TABASCO) to better separate clean samples from noisy samples, especially for the tail classes. TABASCO consists of two new separation metrics that complement each other to compensate for the limitation of using a single metric in sample separation. Extensive experiments on benchmarks demonstrate the effectiveness of our method. Our code is available at https://github.com/Wakings/TABASCO.

Low-light photography produces images with low signal-to-noise ratios due to limited photons. In such conditions, common approximations like the Gaussian noise model fall short, and many denoising techniques fail to remove noise effectively. Although deep-learning methods perform well, they require large datasets of paired images that are impractical to acquire. As a remedy, synthesizing realistic low-light noise has gained significant attention. In this paper, we investigate the ability of diffusion models to capture the complex distribution of low-light noise. We show that a naive application of conventional diffusion models is inadequate for this task and propose three key adaptations that enable high-precision noise generation without calibration or post-processing: a two-branch architecture to better model signal-dependent and signal-independent noise, the incorporation of positional information to capture fixed-pattern noise, and a tailored diffusion noise schedule. Consequently, our model enables the generation of large datasets for training low-light denoising networks, leading to state-of-the-art performance. Through comprehensive analysis, including statistical evaluation and noise decomposition, we provide deeper insights into the characteristics of the generated data.

We present a SemiEmpirical Theory of Learning (SETOL) that explains the remarkable performance of State-Of-The-Art (SOTA) Neural Networks (NNs). We provide a formal explanation of the origin of the fundamental quantities in the phenomenological theory of Heavy-Tailed Self-Regularization (HTSR): the heavy-tailed power-law layer quality metrics, alpha and alpha-hat. In prior work, these metrics have been shown to predict trends in the test accuracies of pretrained SOTA NN models, importantly, without needing access to either testing or training data. Our SETOL uses techniques from statistical mechanics as well as advanced methods from random matrix theory and quantum chemistry. The derivation suggests new mathematical preconditions for ideal learning, including a new metric, ERG, which is equivalent to applying a single step of the Wilson Exact Renormalization Group. We test the assumptions and predictions of SETOL on a simple 3-layer multilayer perceptron (MLP), demonstrating excellent agreement with the key theoretical assumptions. For SOTA NN models, we show how to estimate the individual layer qualities of a trained NN by simply computing the empirical spectral density (ESD) of the layer weight matrices and plugging this ESD into our SETOL formulas. Notably, we examine the performance of the HTSR alpha and the SETOL ERG layer quality metrics, and find that they align remarkably well, both on our MLP and on SOTA NNs.

Diffusion-based image super-resolution methods have demonstrated significant advantages over GAN-based approaches, particularly in terms of perceptual quality. Building upon a lengthy Markov chain, diffusion-based methods possess remarkable modeling capacity, enabling them to achieve outstanding performance in real-world scenarios. Unlike previous methods that focus on modifying the noise schedule or sampling process to enhance performance, our approach emphasizes the improved utilization of LR information. We find that different regions of the LR image can be viewed as corresponding to different timesteps in a diffusion process, where flat areas are closer to the target HR distribution but edge and texture regions are farther away. In these flat areas, applying a slight noise is more advantageous for the reconstruction. We associate this characteristic with uncertainty and propose to apply uncertainty estimate to guide region-specific noise level control, a technique we refer to as Uncertainty-guided Noise Weighting. Pixels with lower uncertainty (i.e., flat regions) receive reduced noise to preserve more LR information, therefore improving performance. Furthermore, we modify the network architecture of previous methods to develop our Uncertainty-guided Perturbation Super-Resolution (UPSR) model. Extensive experimental results demonstrate that, despite reduced model size and training overhead, the proposed UWSR method outperforms current state-of-the-art methods across various datasets, both quantitatively and qualitatively.

Diffusion models excel in generating high-quality images. However, current diffusion models struggle to produce reliable images without guidance methods, such as classifier-free guidance (CFG). Are guidance methods truly necessary? Observing that noise obtained via diffusion inversion can reconstruct high-quality images without guidance, we focus on the initial noise of the denoising pipeline. By mapping Gaussian noise to `guidance-free noise', we uncover that small low-magnitude low-frequency components significantly enhance the denoising process, removing the need for guidance and thus improving both inference throughput and memory. Expanding on this, we propose \ours, a novel method that replaces guidance methods with a single refinement of the initial noise. This refined noise enables high-quality image generation without guidance, within the same diffusion pipeline. Our noise-refining model leverages efficient noise-space learning, achieving rapid convergence and strong performance with just 50K text-image pairs. We validate its effectiveness across diverse metrics and analyze how refined noise can eliminate the need for guidance. See our project page: https://cvlab-kaist.github.io/NoiseRefine/.

Existing long-tailed recognition methods, aiming to train class-balanced models from long-tailed data, generally assume the models would be evaluated on the uniform test class distribution. However, practical test class distributions often violate this assumption (e.g., being either long-tailed or even inversely long-tailed), which may lead existing methods to fail in real applications. In this paper, we study a more practical yet challenging task, called test-agnostic long-tailed recognition, where the training class distribution is long-tailed while the test class distribution is agnostic and not necessarily uniform. In addition to the issue of class imbalance, this task poses another challenge: the class distribution shift between the training and test data is unknown. To tackle this task, we propose a novel approach, called Self-supervised Aggregation of Diverse Experts, which consists of two strategies: (i) a new skill-diverse expert learning strategy that trains multiple experts from a single and stationary long-tailed dataset to separately handle different class distributions; (ii) a novel test-time expert aggregation strategy that leverages self-supervision to aggregate the learned multiple experts for handling unknown test class distributions. We theoretically show that our self-supervised strategy has a provable ability to simulate test-agnostic class distributions. Promising empirical results demonstrate the effectiveness of our method on both vanilla and test-agnostic long-tailed recognition. Code is available at https://github.com/Vanint/SADE-AgnosticLT.

Electrocardiography (ECG) signals are often degraded by noise, which complicates diagnosis in clinical and wearable settings. This study proposes a diffusion-based framework for ECG noise quantification via reconstruction-based anomaly detection, addressing annotation inconsistencies and the limited generalizability of conventional methods. We introduce a distributional evaluation using the Wasserstein-1 distance (W_1), comparing the reconstruction error distributions between clean and noisy ECGs to mitigate inconsistent annotations. Our final model achieved robust noise quantification using only three reverse diffusion steps. The model recorded a macro-average W_1 score of 1.308 across the benchmarks, outperforming the next-best method by over 48%. External validations demonstrated strong generalizability, supporting the exclusion of low-quality segments to enhance diagnostic accuracy and enable timely clinical responses to signal degradation. The proposed method enhances clinical decision-making, diagnostic accuracy, and real-time ECG monitoring capabilities, supporting future advancements in clinical and wearable ECG applications.

The friction coefficient of a particle can depend on its position as it does when the particle is near a wall. We formulate the dynamics of particles with such state-dependent friction coefficients in terms of a general Langevin equation with multiplicative noise, whose evaluation requires the introduction of specific rules. Two common conventions, the Ito and the Stratonovich, provide alternative rules for evaluation of the noise, but other conventions are possible. We show the requirement that a particle's distribution function approach the Boltzmann distribution at long times dictates that a drift term must be added to the Langevin equation. This drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise. We explore the consequences of this result in a number examples with spatially varying diffusion coefficients. We also derive path integral representations for arbitrary interpretation of the noise, and use it in a perturbative study of correlations in a simple system.

Class-wise characteristics of training examples affect the performance of deep classifiers. A well-studied example is when the number of training examples of classes follows a long-tailed distribution, a situation that is likely to yield sub-optimal performance for under-represented classes. This class imbalance problem is conventionally addressed by approaches relying on the class-wise cardinality of training examples, such as data resampling. In this paper, we demonstrate that considering solely the cardinality of classes does not cover all issues causing class imbalance. To measure class imbalance, we propose "Class Uncertainty" as the average predictive uncertainty of the training examples, and we show that this novel measure captures the differences across classes better than cardinality. We also curate SVCI-20 as a novel dataset in which the classes have equal number of training examples but they differ in terms of their hardness; thereby causing a type of class imbalance which cannot be addressed by the approaches relying on cardinality. We incorporate our "Class Uncertainty" measure into a diverse set of ten class imbalance mitigation methods to demonstrate its effectiveness on long-tailed datasets as well as on our SVCI-20. Code and datasets will be made available.

When training neural networks, it has been widely observed that a large step size is essential in stochastic gradient descent (SGD) for obtaining superior models. However, the effect of large step sizes on the success of SGD is not well understood theoretically. Several previous works have attributed this success to the stochastic noise present in SGD. However, we show through a novel set of experiments that the stochastic noise is not sufficient to explain good non-convex training, and that instead the effect of a large learning rate itself is essential for obtaining best performance.We demonstrate the same effects also in the noise-less case, i.e. for full-batch GD. We formally prove that GD with large step size -- on certain non-convex function classes -- follows a different trajectory than GD with a small step size, which can lead to convergence to a global minimum instead of a local one. Our settings provide a framework for future analysis which allows comparing algorithms based on behaviors that can not be observed in the traditional settings.

We study the robustness of classifiers to various kinds of random noise models. In particular, we consider noise drawn uniformly from the ell_p ball for p in [1, infty] and Gaussian noise with an arbitrary covariance matrix. We characterize this robustness to random noise in terms of the distance to the decision boundary of the classifier. This analysis applies to linear classifiers as well as classifiers with locally approximately flat decision boundaries, a condition which is satisfied by state-of-the-art deep neural networks. The predicted robustness is verified experimentally.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size n is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in R^K, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a ell_2 distance of at most varepsilon from the true simplex (for any varepsilon>0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have ngeleft(K^2/varepsilon^2right)e^{Omegaleft(K/SNR^2right)} samples, where SNR stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as SNRgeOmegaleft(K^{1/2}right), the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in ashtiani2018nearly, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

State-of-the-art results on image recognition tasks are achieved using over-parameterized learning algorithms that (nearly) perfectly fit the training set and are known to fit well even random labels. This tendency to memorize the labels of the training data is not explained by existing theoretical analyses. Memorization of the training data also presents significant privacy risks when the training data contains sensitive personal information and thus it is important to understand whether such memorization is necessary for accurate learning. We provide the first conceptual explanation and a theoretical model for this phenomenon. Specifically, we demonstrate that for natural data distributions memorization of labels is necessary for achieving close-to-optimal generalization error. Crucially, even labels of outliers and noisy labels need to be memorized. The model is motivated and supported by the results of several recent empirical works. In our model, data is sampled from a mixture of subpopulations and our results show that memorization is necessary whenever the distribution of subpopulation frequencies is long-tailed. Image and text data is known to be long-tailed and therefore our results establish a formal link between these empirical phenomena. Our results allow to quantify the cost of limiting memorization in learning and explain the disparate effects that privacy and model compression have on different subgroups.

Image interpolation based on diffusion models is promising in creating fresh and interesting images. Advanced interpolation methods mainly focus on spherical linear interpolation, where images are encoded into the noise space and then interpolated for denoising to images. However, existing methods face challenges in effectively interpolating natural images (not generated by diffusion models), thereby restricting their practical applicability. Our experimental investigations reveal that these challenges stem from the invalidity of the encoding noise, which may no longer obey the expected noise distribution, e.g., a normal distribution. To address these challenges, we propose a novel approach to correct noise for image interpolation, NoiseDiffusion. Specifically, NoiseDiffusion approaches the invalid noise to the expected distribution by introducing subtle Gaussian noise and introduces a constraint to suppress noise with extreme values. In this context, promoting noise validity contributes to mitigating image artifacts, but the constraint and introduced exogenous noise typically lead to a reduction in signal-to-noise ratio, i.e., loss of original image information. Hence, NoiseDiffusion performs interpolation within the noisy image space and injects raw images into these noisy counterparts to address the challenge of information loss. Consequently, NoiseDiffusion enables us to interpolate natural images without causing artifacts or information loss, thus achieving the best interpolation results.

Deep long-tailed learning, one of the most challenging problems in visual recognition, aims to train well-performing deep models from a large number of images that follow a long-tailed class distribution. In the last decade, deep learning has emerged as a powerful recognition model for learning high-quality image representations and has led to remarkable breakthroughs in generic visual recognition. However, long-tailed class imbalance, a common problem in practical visual recognition tasks, often limits the practicality of deep network based recognition models in real-world applications, since they can be easily biased towards dominant classes and perform poorly on tail classes. To address this problem, a large number of studies have been conducted in recent years, making promising progress in the field of deep long-tailed learning. Considering the rapid evolution of this field, this paper aims to provide a comprehensive survey on recent advances in deep long-tailed learning. To be specific, we group existing deep long-tailed learning studies into three main categories (i.e., class re-balancing, information augmentation and module improvement), and review these methods following this taxonomy in detail. Afterward, we empirically analyze several state-of-the-art methods by evaluating to what extent they address the issue of class imbalance via a newly proposed evaluation metric, i.e., relative accuracy. We conclude the survey by highlighting important applications of deep long-tailed learning and identifying several promising directions for future research.

Significant work has been recently dedicated to the stochastic delayed bandit setting because of its relevance in applications. The applicability of existing algorithms is however restricted by the fact that strong assumptions are often made on the delay distributions, such as full observability, restrictive shape constraints, or uniformity over arms. In this work, we weaken them significantly and only assume that there is a bound on the tail of the delay. In particular, we cover the important case where the delay distributions vary across arms, and the case where the delays are heavy-tailed. Addressing these difficulties, we propose a simple but efficient UCB-based algorithm called the PatientBandits. We provide both problems-dependent and problems-independent bounds on the regret as well as performance lower bounds.

Deep learning weather models now match numerical weather prediction accuracy while running orders of magnitude faster, but produce deterministic forecasts without uncertainty estimates, a critical gap for high-stakes decisions during extreme weather events. This paper proposes Neural Tangent Kernel-based uncertainty quantification (NTK-UQ) using last-layer empirical features. Theoretical analysis predicts that UQ quality is architecture-dependent through two mechanisms. First, a variance collapse mechanism explains when UQ fails: when the eigenvalue truncation rank approaches the effective rank of the feature space, the GP correction term consumes nearly all prior variance, destroying discrimination between tropical cyclones and routine conditions; architectures with concentrated spectra (spectral operators) require aggressive truncation (k leq 10), while attention-based models tolerate full-rank computation. Second, decomposition performance depends on the non-Gaussian, heavy-tailed structure of extreme weather: Independent Component Analysis exploits higher-order statistics (kurtosis, negentropy) to isolate heavy-tailed extreme-event features, achieving higher discrimination than singular value decomposition, which captures only second-order variance. A data-driven selection rule chooses ICA or SVD from the feature eigenspectrum concentration ratio, correctly prescribing the superior decomposition for all four evaluated architectures. Compared to split conformal prediction (the natural post-hoc baseline), NTK-UQ achieves 31--37\% sharper prediction intervals at 90\% coverage, and uniquely produces adaptive intervals that scale with extreme event severity, which conformal prediction cannot achieve by construction. The framework requires no retraining; inference-time uncertainty requires only a single matrix-vector product per sample.

Retrieval-Augmented Generation (RAG) has emerged as a crucial method for addressing hallucinations in large language models (LLMs). While recent research has extended RAG models to complex noisy scenarios, these explorations often confine themselves to limited noise types and presuppose that noise is inherently detrimental to LLMs, potentially deviating from real-world retrieval environments and restricting practical applicability. In this paper, we define seven distinct noise types from a linguistic perspective and establish a Noise RAG Benchmark (NoiserBench), a comprehensive evaluation framework encompassing multiple datasets and reasoning tasks. Through empirical evaluation of eight representative LLMs with diverse architectures and scales, we reveal that these noises can be further categorized into two practical groups: noise that is beneficial to LLMs (aka beneficial noise) and noise that is harmful to LLMs (aka harmful noise). While harmful noise generally impairs performance, beneficial noise may enhance several aspects of model capabilities and overall performance. Our analysis offers insights for developing more robust, adaptable RAG solutions and mitigating hallucinations across diverse retrieval scenarios.

Recognition problems in long-tailed data, in which the sample size per class is heavily skewed, have gained importance because the distribution of the sample size per class in a dataset is generally exponential unless the sample size is intentionally adjusted. Various methods have been devised to address these problems. Recently, weight balancing, which combines well-known classical regularization techniques with two-stage training, has been proposed. Despite its simplicity, it is known for its high performance compared with existing methods devised in various ways. However, there is a lack of understanding as to why this method is effective for long-tailed data. In this study, we analyze weight balancing by focusing on neural collapse and the cone effect at each training stage and found that it can be decomposed into an increase in Fisher's discriminant ratio of the feature extractor caused by weight decay and cross entropy loss and implicit logit adjustment caused by weight decay and class-balanced loss. Our analysis enables the training method to be further simplified by reducing the number of training stages to one while increasing accuracy.

Long-tailed data distributions pose challenges for a variety of domains like e-commerce, finance, biomedical science, and cyber security, where the performance of machine learning models is often dominated by head categories while tail categories are inadequately learned. This work aims to provide a systematic view of long-tailed learning with regard to three pivotal angles: (A1) the characterization of data long-tailedness, (A2) the data complexity of various domains, and (A3) the heterogeneity of emerging tasks. We develop HeroLT, a comprehensive long-tailed learning benchmark integrating 18 state-of-the-art algorithms, 10 evaluation metrics, and 17 real-world datasets across 6 tasks and 4 data modalities. HeroLT with novel angles and extensive experiments (315 in total) enables effective and fair evaluation of newly proposed methods compared with existing baselines on varying dataset types. Finally, we conclude by highlighting the significant applications of long-tailed learning and identifying several promising future directions. For accessibility and reproducibility, we open-source our benchmark HeroLT and corresponding results at https://github.com/SSSKJ/HeroLT.

The proliferation of probabilistic AI has promoted proposals for specialized stochastic computers. Despite promising efficiency gains, these proposals have failed to gain traction because they rely on fundamentally limited modeling techniques and exotic, unscalable hardware. In this work, we address these shortcomings by proposing an all-transistor probabilistic computer that implements powerful denoising models at the hardware level. A system-level analysis indicates that devices based on our architecture could achieve performance parity with GPUs on a simple image benchmark using approximately 10,000 times less energy.

Detecting oscillations in solar and stellar time series is complicated by non-stationary red noise and evolving background emission. Methods based on detrending and AR(1)-based wavelet analysis can introduce spurious periodicities and do not adequately describe time-dependent backgrounds. We develop a Bayesian approach that combines the continuous wavelet transform with MCMC sampling to infer a time-dependent background spectrum. The background is represented by a power-law plus white-noise component, with parameters allowed to vary smoothly in time, so that significance levels can be evaluated locally without explicit detrending. Tests with synthetic data show that injected oscillations are recovered reliably, while false detections are suppressed in pure-noise cases. Using a frequency-domain signal-to-noise ratio (S/N), we find that oscillations can be identified robustly when the S/N is greater than or equal to 2 under mixed noise conditions. The detectable period range is limited by wavelet resolution, from about 3-4 sampling intervals up to roughly one-quarter of the total duration. Application to GOES soft X-ray flare observations shows that the method isolates quasi-periodic oscillations with improved temporal localization compared to standard wavelet and Fourier-based approaches. Meanwhile, this behavior is consistent across a range of noise conditions and signal morphologies.

Supervised fine-tuning (SFT) plays a crucial role in adapting large language models (LLMs) to specific domains or tasks. However, as demonstrated by empirical experiments, the collected data inevitably contains noise in practical applications, which poses significant challenges to model performance on downstream tasks. Therefore, there is an urgent need for a noise-robust SFT framework to enhance model capabilities in downstream tasks. To address this challenge, we introduce a robust SFT framework (RobustFT) that performs noise detection and relabeling on downstream task data. For noise identification, our approach employs a multi-expert collaborative system with inference-enhanced models to achieve superior noise detection. In the denoising phase, we utilize a context-enhanced strategy, which incorporates the most relevant and confident knowledge followed by careful assessment to generate reliable annotations. Additionally, we introduce an effective data selection mechanism based on response entropy, ensuring only high-quality samples are retained for fine-tuning. Extensive experiments conducted on multiple LLMs across five datasets demonstrate RobustFT's exceptional performance in noisy scenarios.

The all-sky 408 MHz map of Haslam et al. is one the most important total-power radio surveys. It has been widely used to study diffuse synchrotron radiation from our Galaxy and as a template to remove foregrounds in cosmic microwave background data. However, there are a number of issues associated with it that must be dealt with, including large-scale striations and contamination from extragalactic radio sources. We have re-evaluated and re-processed the rawest data available to produce a new and improved 408 MHz all-sky map. We first quantify the positional accuracy (approx 7 arcmin) and effective beam (56.0pm1.0 arcmin) of the four individual surveys from which it was assembled. Large-scale striations associated with 1/f noise in the scan direction are reduced to a level ll 1 K using a Fourier-based filtering technique. The most important improvement results from the removal of extragalactic sources. We have used an iterative combination of two techniques -- two-dimensional Gaussian fitting and minimum curvature spline surface inpainting -- to remove the brightest sources (gtrsim 2 Jy), which provides a significant improvement over previous versions of the map. We quantify the impact with power spectra and a template fitting analysis of foregrounds to the WMAP data. The new map is publicly available and is recommended as the template of choice for large-scale diffuse Galactic synchrotron emission. We also provide a higher resolution map with small-scale fluctuations added, assuming a power-law angular power spectrum down to the pixel scale (1.7 arcmin). This should prove useful in simulations used for studying the feasibility of detecting HI fluctuations from the Epoch of Reionization.

Poisson noise suppression is an important preprocessing step in several applications, such as medical imaging, microscopy, and astronomical imaging. In this work, we propose a novel patch-wise Poisson noise removal strategy, in which the MMSE estimator is utilized in order to produce the denoising result for each image patch. Fast and accurate computation of the MMSE estimator is carried out using k-d tree search followed by search in the K-nearest neighbor graph. Our experiments show that the proposed method is the preferable choice for low signal-to-noise ratios.

We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a symmetry conjecture that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to 90% narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.

While deep convolutional neural networks (CNNs) have achieved impressive success in image denoising with additive white Gaussian noise (AWGN), their performance remains limited on real-world noisy photographs. The main reason is that their learned models are easy to overfit on the simplified AWGN model which deviates severely from the complicated real-world noise model. In order to improve the generalization ability of deep CNN denoisers, we suggest training a convolutional blind denoising network (CBDNet) with more realistic noise model and real-world noisy-clean image pairs. On the one hand, both signal-dependent noise and in-camera signal processing pipeline is considered to synthesize realistic noisy images. On the other hand, real-world noisy photographs and their nearly noise-free counterparts are also included to train our CBDNet. To further provide an interactive strategy to rectify denoising result conveniently, a noise estimation subnetwork with asymmetric learning to suppress under-estimation of noise level is embedded into CBDNet. Extensive experimental results on three datasets of real-world noisy photographs clearly demonstrate the superior performance of CBDNet over state-of-the-arts in terms of quantitative metrics and visual quality. The code has been made available at https://github.com/GuoShi28/CBDNet.

Speckle is a multiplicative noise which affects all coherent imaging modalities including Synthetic Aperture Radar (SAR) images. The presence of speckle degrades the image quality and adversely affects the performance of SAR image understanding applications such as automatic target recognition and change detection. Thus, SAR despeckling is an important problem in remote sensing. In this paper, we introduce SAR-DDPM, a denoising diffusion probabilistic model for SAR despeckling. The proposed method comprises of a Markov chain that transforms clean images to white Gaussian noise by repeatedly adding random noise. The despeckled image is recovered by a reverse process which iteratively predicts the added noise using a noise predictor which is conditioned on the speckled image. In addition, we propose a new inference strategy based on cycle spinning to improve the despeckling performance. Our experiments on both synthetic and real SAR images demonstrate that the proposed method achieves significant improvements in both quantitative and qualitative results over the state-of-the-art despeckling methods.

It is well known that many open-released foundational diffusion models have difficulty in generating images that substantially depart from average brightness, despite such images being present in the training data. This is due to an inconsistency: while denoising starts from pure Gaussian noise during inference, the training noise schedule retains residual data even in the final timestep distribution, due to difficulties in numerical conditioning in mainstream formulation, leading to unintended bias during inference. To mitigate this issue, certain epsilon-prediction models are combined with an ad-hoc offset-noise methodology. In parallel, some contemporary models have adopted zero-terminal SNR noise schedules together with v-prediction, which necessitate major alterations to pre-trained models. However, such changes risk destabilizing a large multitude of community-driven applications anchored on these pre-trained models. In light of this, our investigation revisits the fundamental causes, leading to our proposal of an innovative and principled remedy, called One More Step (OMS). By integrating a compact network and incorporating an additional simple yet effective step during inference, OMS elevates image fidelity and harmonizes the dichotomy between training and inference, while preserving original model parameters. Once trained, various pre-trained diffusion models with the same latent domain can share the same OMS module.

We present the Noisy Ostracods, a noisy dataset for genus and species classification of crustacean ostracods with specialists' annotations. Over the 71466 specimens collected, 5.58% of them are estimated to be noisy (possibly problematic) at genus level. The dataset is created to addressing a real-world challenge: creating a clean fine-grained taxonomy dataset. The Noisy Ostracods dataset has diverse noises from multiple sources. Firstly, the noise is open-set, including new classes discovered during curation that were not part of the original annotation. The dataset has pseudo-classes, where annotators misclassified samples that should belong to an existing class into a new pseudo-class. The Noisy Ostracods dataset is highly imbalanced with a imbalance factor ρ = 22429. This presents a unique challenge for robust machine learning methods, as existing approaches have not been extensively evaluated on fine-grained classification tasks with such diverse real-world noise. Initial experiments using current robust learning techniques have not yielded significant performance improvements on the Noisy Ostracods dataset compared to cross-entropy training on the raw, noisy data. On the other hand, noise detection methods have underperformed in error hit rate compared to naive cross-validation ensembling for identifying problematic labels. These findings suggest that the fine-grained, imbalanced nature, and complex noise characteristics of the dataset present considerable challenges for existing noise-robust algorithms. By openly releasing the Noisy Ostracods dataset, our goal is to encourage further research into the development of noise-resilient machine learning methods capable of effectively handling diverse, real-world noise in fine-grained classification tasks. The dataset, along with its evaluation protocols, can be accessed at https://github.com/H-Jamieu/Noisy_ostracods.

We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of 1-bit quantization -- allow a high accuracy approximation of the underlying kernels, and the approximation error decays at least polynomially fast as the dimension of the RFFs increases. We also show that the quantized RFFs can be further compressed, yielding an excellent trade-off between memory use and accuracy. Namely, the approximation error now decays exponentially as a function of the bits used. Moreover, we empirically show by testing the performance of our methods on several machine learning tasks that our method compares favorably to other state of the art quantization methods in this context.

Denoising diffusion probabilistic models (DDPMs) employ a sequence of white Gaussian noise samples to generate an image. In analogy with GANs, those noise maps could be considered as the latent code associated with the generated image. However, this native noise space does not possess a convenient structure, and is thus challenging to work with in editing tasks. Here, we propose an alternative latent noise space for DDPM that enables a wide range of editing operations via simple means, and present an inversion method for extracting these edit-friendly noise maps for any given image (real or synthetically generated). As opposed to the native DDPM noise space, the edit-friendly noise maps do not have a standard normal distribution and are not statistically independent across timesteps. However, they allow perfect reconstruction of any desired image, and simple transformations on them translate into meaningful manipulations of the output image (e.g., shifting, color edits). Moreover, in text-conditional models, fixing those noise maps while changing the text prompt, modifies semantics while retaining structure. We illustrate how this property enables text-based editing of real images via the diverse DDPM sampling scheme (in contrast to the popular non-diverse DDIM inversion). We also show how it can be used within existing diffusion-based editing methods to improve their quality and diversity.

Image denoising is a fundamental challenge in computer vision, with applications in photography and medical imaging. While deep learning-based methods have shown remarkable success, their reliance on specific noise distributions limits generalization to unseen noise types and levels. Existing approaches attempt to address this with extensive training data and high computational resources but they still suffer from overfitting. To address these issues, we conduct image denoising by utilizing dynamically generated kernels via efficient operations. This approach helps prevent overfitting and improves resilience to unseen noise. Specifically, our method leverages a Feature Extraction Module for robust noise-invariant features, Global Statistics and Local Correlation Modules to capture comprehensive noise characteristics and structural correlations. The Kernel Prediction Module then employs these cues to produce pixel-wise varying kernels adapted to local structures, which are then applied iteratively for denoising. This ensures both efficiency and superior restoration quality. Despite being trained on single-level Gaussian noise, our compact model (~ 0.04 M) excels across diverse noise types and levels, demonstrating the promise of iterative dynamic filtering for practical image denoising.

As the class size grows, maintaining a balanced dataset across many classes is challenging because the data are long-tailed in nature; it is even impossible when the sample-of-interest co-exists with each other in one collectable unit, e.g., multiple visual instances in one image. Therefore, long-tailed classification is the key to deep learning at scale. However, existing methods are mainly based on re-weighting/re-sampling heuristics that lack a fundamental theory. In this paper, we establish a causal inference framework, which not only unravels the whys of previous methods, but also derives a new principled solution. Specifically, our theory shows that the SGD momentum is essentially a confounder in long-tailed classification. On one hand, it has a harmful causal effect that misleads the tail prediction biased towards the head. On the other hand, its induced mediation also benefits the representation learning and head prediction. Our framework elegantly disentangles the paradoxical effects of the momentum, by pursuing the direct causal effect caused by an input sample. In particular, we use causal intervention in training, and counterfactual reasoning in inference, to remove the "bad" while keep the "good". We achieve new state-of-the-arts on three long-tailed visual recognition benchmarks: Long-tailed CIFAR-10/-100, ImageNet-LT for image classification and LVIS for instance segmentation.

Reconstruction-based methods are a dominant paradigm in time series anomaly detection (TSAD), however, their near-universal reliance on Mean Squared Error (MSE) loss results in statistically flawed reconstruction residuals. This fundamental weakness leads to noisy, unstable anomaly scores, hindering reliable detection. To address this, we propose Constrained Gaussian-Noise Optimization and Smoothing (COGNOS), a universal, model-agnostic enhancement framework that tackles this issue at its source. COGNOS introduces a novel Gaussian-White Noise Regularization strategy during training, which directly constrains the model's output residuals to conform to a Gaussian white noise distribution. This engineered statistical property creates the ideal precondition for our second contribution: Adaptive Residual Kalman Smoother that operates as a statistically robust estimator to denoise the raw anomaly scores. Extensive experiments on multiple benchmarks demonstrate that COGNOS consistently enhances the performance of state-of-the-art backbones significantly, validating the efficacy of coupling statistical regularization with adaptive filtering.

Long-tailed learning has garnered increasing attention due to its practical significance. Among the various approaches, the fine-tuning paradigm has gained considerable interest with the advent of foundation models. However, most existing methods primarily focus on leveraging knowledge from these models, overlooking the inherent biases introduced by the imbalanced training data they rely on. In this paper, we examine how such imbalances from pre-training affect long-tailed downstream tasks. Specifically, we find the imbalance biases inherited in foundation models on downstream task as parameter imbalance and data imbalance. During fine-tuning, we observe that parameter imbalance plays a more critical role, while data imbalance can be mitigated using existing re-balancing strategies. Moreover, we find that parameter imbalance cannot be effectively addressed by current re-balancing techniques, such as adjusting the logits, during training, unlike data imbalance. To tackle both imbalances simultaneously, we build our method on causal learning and view the incomplete semantic factor as the confounder, which brings spurious correlations between input samples and labels. To resolve the negative effects of this, we propose a novel backdoor adjustment method that learns the true causal effect between input samples and labels, rather than merely fitting the correlations in the data. Notably, we achieve an average performance increase of about 1.67% on each dataset.

The pre-trained text-to-image diffusion models have been increasingly employed to tackle the real-world image super-resolution (Real-ISR) problem due to their powerful generative image priors. Most of the existing methods start from random noise to reconstruct the high-quality (HQ) image under the guidance of the given low-quality (LQ) image. While promising results have been achieved, such Real-ISR methods require multiple diffusion steps to reproduce the HQ image, increasing the computational cost. Meanwhile, the random noise introduces uncertainty in the output, which is unfriendly to image restoration tasks. To address these issues, we propose a one-step effective diffusion network, namely OSEDiff, for the Real-ISR problem. We argue that the LQ image contains rich information to restore its HQ counterpart, and hence the given LQ image can be directly taken as the starting point for diffusion, eliminating the uncertainty introduced by random noise sampling. We finetune the pre-trained diffusion network with trainable layers to adapt it to complex image degradations. To ensure that the one-step diffusion model could yield HQ Real-ISR output, we apply variational score distillation in the latent space to conduct KL-divergence regularization. As a result, our OSEDiff model can efficiently and effectively generate HQ images in just one diffusion step. Our experiments demonstrate that OSEDiff achieves comparable or even better Real-ISR results, in terms of both objective metrics and subjective evaluations, than previous diffusion model-based Real-ISR methods that require dozens or hundreds of steps. The source codes are released at https://github.com/cswry/OSEDiff.

Text-to-image diffusion models trained on a fixed set of resolutions often fail to generalize, even when asked to generate images at lower resolutions than those seen during training. High-resolution text-to-image generators are currently unable to easily offer an out-of-the-box budget-efficient alternative to their users who might not need high-resolution images. We identify a key technical insight in diffusion models that when addressed can help tackle this limitation: Noise schedulers have unequal perceptual effects across resolutions. The same level of noise removes disproportionately more signal from lower-resolution images than from high-resolution images, leading to a train-test mismatch. We propose NoiseShift, a training-free method that recalibrates the noise level of the denoiser conditioned on resolution size. NoiseShift requires no changes to model architecture or sampling schedule and is compatible with existing models. When applied to Stable Diffusion 3, Stable Diffusion 3.5, and Flux-Dev, quality at low resolutions is significantly improved. On LAION-COCO, NoiseShift improves SD3.5 by 15.89%, SD3 by 8.56%, and Flux-Dev by 2.44% in FID on average. On CelebA, NoiseShift improves SD3.5 by 10.36%, SD3 by 5.19%, and Flux-Dev by 3.02% in FID on average. These results demonstrate the effectiveness of NoiseShift in mitigating resolution-dependent artifacts and enhancing the quality of low-resolution image generation.

Diffusion models have emerged as the de facto choice for generating visual signals. However, training a single model to predict noise across various levels poses significant challenges, necessitating numerous iterations and incurring significant computational costs. Various approaches, such as loss weighting strategy design and architectural refinements, have been introduced to expedite convergence. In this study, we propose a novel approach to design the noise schedule for enhancing the training of diffusion models. Our key insight is that the importance sampling of the logarithm of the Signal-to-Noise ratio (logSNR), theoretically equivalent to a modified noise schedule, is particularly beneficial for training efficiency when increasing the sample frequency around log SNR=0. We empirically demonstrate the superiority of our noise schedule over the standard cosine schedule. Furthermore, we highlight the advantages of our noise schedule design on the ImageNet benchmark, showing that the designed schedule consistently benefits different prediction targets.

Recorrupted-to-Recorrupted (R2R) has emerged as a methodology for training deep networks for image restoration in a self-supervised manner from noisy measurement data alone, demonstrating equivalence in expectation to the supervised squared loss in the case of Gaussian noise. However, its effectiveness with non-Gaussian noise remains unexplored. In this paper, we propose Generalized R2R (GR2R), extending the R2R framework to handle a broader class of noise distribution as additive noise like log-Rayleigh and address the natural exponential family including Poisson and Gamma noise distributions, which play a key role in many applications including low-photon imaging and synthetic aperture radar. We show that the GR2R loss is an unbiased estimator of the supervised loss and that the popular Stein's unbiased risk estimator can be seen as a special case. A series of experiments with Gaussian, Poisson, and Gamma noise validate GR2R's performance, showing its effectiveness compared to other self-supervised methods.

Normalizing flows, a popular class of deep generative models, often fail to represent extreme phenomena observed in real-world processes. In particular, existing normalizing flow architectures struggle to model multivariate extremes, characterized by heavy-tailed marginal distributions and asymmetric tail dependence among variables. In light of this shortcoming, we propose COMET (COpula Multivariate ExTreme) Flows, which decompose the process of modeling a joint distribution into two parts: (i) modeling its marginal distributions, and (ii) modeling its copula distribution. COMET Flows capture heavy-tailed marginal distributions by combining a parametric tail belief at extreme quantiles of the marginals with an empirical kernel density function at mid-quantiles. In addition, COMET Flows capture asymmetric tail dependence among multivariate extremes by viewing such dependence as inducing a low-dimensional manifold structure in feature space. Experimental results on both synthetic and real-world datasets demonstrate the effectiveness of COMET Flows in capturing both heavy-tailed marginals and asymmetric tail dependence compared to other state-of-the-art baseline architectures. All code is available on GitHub at https://github.com/andrewmcdonald27/COMETFlows.

Denoising diffusion models have recently shown impressive results in generative tasks. By learning powerful priors from huge collections of training images, such models are able to gradually modify complete noise to a clean natural image via a sequence of small denoising steps, seemingly making them well-suited for single image denoising. However, effectively applying denoising diffusion models to removal of realistic noise is more challenging than it may seem, since their formulation is based on additive white Gaussian noise, unlike noise in real-world images. In this work, we present SVNR, a novel formulation of denoising diffusion that assumes a more realistic, spatially-variant noise model. SVNR enables using the noisy input image as the starting point for the denoising diffusion process, in addition to conditioning the process on it. To this end, we adapt the diffusion process to allow each pixel to have its own time embedding, and propose training and inference schemes that support spatially-varying time maps. Our formulation also accounts for the correlation that exists between the condition image and the samples along the modified diffusion process. In our experiments we demonstrate the advantages of our approach over a strong diffusion model baseline, as well as over a state-of-the-art single image denoising method.

Recent advances in foundation models have emphasized the need to align pre-trained models with specialized domains using small, curated datasets. Studies on these foundation models underscore the importance of low-data training and fine-tuning. This topic, well-known in natural language processing (NLP), has also gained increasing attention in the emerging field of scientific machine learning (SciML). To address the limitations of low-data training and fine-tuning, we draw inspiration from Heavy-Tailed Self-Regularization (HT-SR) theory, analyzing the shape of empirical spectral densities (ESDs) and revealing an imbalance in training quality across different model layers. To mitigate this issue, we adapt a recently proposed layer-wise learning rate scheduler, TempBalance, which effectively balances training quality across layers and enhances low-data training and fine-tuning for both NLP and SciML tasks. Notably, TempBalance demonstrates increasing performance gains as the amount of available tuning data decreases. Comparative analyses further highlight the effectiveness of TempBalance and its adaptability as an "add-on" method for improving model performance.

Deep learning methods for unsupervised registration often rely on objectives that assume a uniform noise level across the spatial domain (e.g. mean-squared error loss), but noise distributions are often heteroscedastic and input-dependent in real-world medical images. Thus, this assumption often leads to degradation in registration performance, mainly due to the undesired influence of noise-induced outliers. To mitigate this, we propose a framework for heteroscedastic image uncertainty estimation that can adaptively reduce the influence of regions with high uncertainty during unsupervised registration. The framework consists of a collaborative training strategy for the displacement and variance estimators, and a novel image fidelity weighting scheme utilizing signal-to-noise ratios. Our approach prevents the model from being driven away by spurious gradients caused by the simplified homoscedastic assumption, leading to more accurate displacement estimation. To illustrate its versatility and effectiveness, we tested our framework on two representative registration architectures across three medical image datasets. Our method consistently outperforms baselines and produces sensible uncertainty estimates. The code is publicly available at https://voldemort108x.github.io/hetero_uncertainty/.

While 2D diffusion models generate realistic, high-detail images, 3D shape generation methods like Score Distillation Sampling (SDS) built on these 2D diffusion models produce cartoon-like, over-smoothed shapes. To help explain this discrepancy, we show that the image guidance used in Score Distillation can be understood as the velocity field of a 2D denoising generative process, up to the choice of a noise term. In particular, after a change of variables, SDS resembles a high-variance version of Denoising Diffusion Implicit Models (DDIM) with a differently-sampled noise term: SDS introduces noise i.i.d. randomly at each step, while DDIM infers it from the previous noise predictions. This excessive variance can lead to over-smoothing and unrealistic outputs. We show that a better noise approximation can be recovered by inverting DDIM in each SDS update step. This modification makes SDS's generative process for 2D images almost identical to DDIM. In 3D, it removes over-smoothing, preserves higher-frequency detail, and brings the generation quality closer to that of 2D samplers. Experimentally, our method achieves better or similar 3D generation quality compared to other state-of-the-art Score Distillation methods, all without training additional neural networks or multi-view supervision, and providing useful insights into relationship between 2D and 3D asset generation with diffusion models.

Real-world data often exhibits long tail distributions with heavy class imbalance, where the majority classes can dominate the training process and alter the decision boundaries of the minority classes. Recently, researchers have investigated the potential of supervised contrastive learning for long-tailed recognition, and demonstrated that it provides a strong performance gain. In this paper, we show that while supervised contrastive learning can help improve performance, past baselines suffer from poor uniformity brought in by imbalanced data distribution. This poor uniformity manifests in samples from the minority class having poor separability in the feature space. To address this problem, we propose targeted supervised contrastive learning (TSC), which improves the uniformity of the feature distribution on the hypersphere. TSC first generates a set of targets uniformly distributed on a hypersphere. It then makes the features of different classes converge to these distinct and uniformly distributed targets during training. This forces all classes, including minority classes, to maintain a uniform distribution in the feature space, improves class boundaries, and provides better generalization even in the presence of long-tail data. Experiments on multiple datasets show that TSC achieves state-of-the-art performance on long-tailed recognition tasks.

Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency. For a fixed stepsize, the iterates typically admit a stationary distribution that is rarely tractable. Prior work shows that as the stepsize αdownarrow 0, the centered-and-scaled steady state converges weakly to a Gaussian random vector. However, for fixed α, this weak convergence offers no usable error bound for approximating the steady-state by its Gaussian limit. This paper provides explicit, non-asymptotic error bounds for fixed α. We first prove general-purpose theorems that bound the Wasserstein distance between the centered-scaled steady state and an appropriate Gaussian distribution, under regularity conditions for drift and moment conditions for noise. To ensure broad applicability, we cover both i.i.d. and Markovian noise models. We then instantiate these theorems for three representative SA settings: (1) stochastic gradient descent (SGD) for smooth strongly convex objectives, (2) linear SA, and (3) contractive nonlinear SA. We obtain dimension- and stepsize-dependent, explicit bounds in Wasserstein distance of order α^{1/2}log(1/α) for small α. Building on the Wasserstein approximation error, we further derive non-uniform Berry--Esseen-type tail bounds that compare the steady-state tail probability to Gaussian tails. We achieve an explicit error term that decays in both the deviation level and stepsize α. We adapt the same analysis for SGD beyond strongly convexity and study general convex objectives. We identify a non-Gaussian (Gibbs) limiting law under the correct scaling, which is validated numerically, and provide a corresponding pre-limit Wasserstein error bound.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula (1-β^{n})/(1-β), where n is the number of samples and βin [0,1) is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.

It is widely believed that noise conditioning is indispensable for denoising diffusion models to work successfully. This work challenges this belief. Motivated by research on blind image denoising, we investigate a variety of denoising-based generative models in the absence of noise conditioning. To our surprise, most models exhibit graceful degradation, and in some cases, they even perform better without noise conditioning. We provide a theoretical analysis of the error caused by removing noise conditioning and demonstrate that our analysis aligns with empirical observations. We further introduce a noise-unconditional model that achieves a competitive FID of 2.23 on CIFAR-10, significantly narrowing the gap to leading noise-conditional models. We hope our findings will inspire the community to revisit the foundations and formulations of denoising generative models.

Internal features from large-scale pre-trained diffusion models have recently been established as powerful semantic descriptors for a wide range of downstream tasks. Works that use these features generally need to add noise to images before passing them through the model to obtain the semantic features, as the models do not offer the most useful features when given images with little to no noise. We show that this noise has a critical impact on the usefulness of these features that cannot be remedied by ensembling with different random noises. We address this issue by introducing a lightweight, unsupervised fine-tuning method that enables diffusion backbones to provide high-quality, noise-free semantic features. We show that these features readily outperform previous diffusion features by a wide margin in a wide variety of extraction setups and downstream tasks, offering better performance than even ensemble-based methods at a fraction of the cost.

Most applications of machine learning to classification assume a closed set of balanced classes. This is at odds with the real world, where class occurrence statistics often follow a long-tailed power-law distribution and it is unlikely that all classes are seen in a single sample. Nonparametric Bayesian models naturally capture this phenomenon, but have significant practical barriers to widespread adoption, namely implementation complexity and computational inefficiency. To address this, we present a method for extracting the inductive bias from a nonparametric Bayesian model and transferring it to an artificial neural network. By simulating data with a nonparametric Bayesian prior, we can metalearn a sequence model that performs inference over an unlimited set of classes. After training, this "neural circuit" has distilled the corresponding inductive bias and can successfully perform sequential inference over an open set of classes. Our experimental results show that the metalearned neural circuit achieves comparable or better performance than particle filter-based methods for inference in these models while being faster and simpler to use than methods that explicitly incorporate Bayesian nonparametric inference.

We derive a minimalist but powerful deterministic denoising-diffusion model. While denoising diffusion has shown great success in many domains, its underlying theory remains largely inaccessible to non-expert users. Indeed, an understanding of graduate-level concepts such as Langevin dynamics or score matching appears to be required to grasp how it works. We propose an alternative approach that requires no more than undergrad calculus and probability. We consider two densities and observe what happens when random samples from these densities are blended (linearly interpolated). We show that iteratively blending and deblending samples produces random paths between the two densities that converge toward a deterministic mapping. This mapping can be evaluated with a neural network trained to deblend samples. We obtain a model that behaves like deterministic denoising diffusion: it iteratively maps samples from one density (e.g., Gaussian noise) to another (e.g., cat images). However, compared to the state-of-the-art alternative, our model is simpler to derive, simpler to implement, more numerically stable, achieves higher quality results in our experiments, and has interesting connections to computer graphics.

We study estimation and clustering in Gaussian mixture models under variance misspecification. Observations are generated with true variance σ^2, while the component means are estimated using a likelihood with variance τ^2, yielding a family of mismatched likelihood functions parameterized by the ratio ρ=τ/σ. We show that the interplay between ρ and the signal-to-noise ratio (SNR) induces a sharp phase diagram. Under correct specification (ρ=1), maximum likelihood recovers the true means, independently of the SNR. However, once the model is misspecified, two different regimes emerge. Under under-smoothing (ρ<1), the estimated Gaussian means are displaced from the truth, and in low SNR this discrepancy grows as the SNR decreases: for every fixed ρ<1, the squared error scales as SNR^{-1}. Under over-smoothing (ρ>1), the fitted likelihood blurs the cluster separation, causing distinct component means to collapse towards the overall mixture center once ρ^2 exceeds a threshold of the form 1 + λ,SNR, where λ depends on the geometry of the true means. We further show that the hard assignment objective arises as the limit τto 0 of the same mismatched likelihood family, and derive corresponding low- and high-SNR results for hard-assignment mean estimation and latent-label recovery. Furthermore, in low SNR, Bayes-optimal clustering is close to random guessing, and the hard-assignment target remains far from the true means. These results show that in low-SNR applications, even mild variance misspecification or hard-assignment procedures can induce substantial bias, whereas in high SNR these effects are largely absent.

In online algorithm selection (OAS), instances of an algorithmic problem class are presented to an agent one after another, and the agent has to quickly select a presumably best algorithm from a fixed set of candidate algorithms. For decision problems such as satisfiability (SAT), quality typically refers to the algorithm's runtime. As the latter is known to exhibit a heavy-tail distribution, an algorithm is normally stopped when exceeding a predefined upper time limit. As a consequence, machine learning methods used to optimize an algorithm selection strategy in a data-driven manner need to deal with right-censored samples, a problem that has received little attention in the literature so far. In this work, we revisit multi-armed bandit algorithms for OAS and discuss their capability of dealing with the problem. Moreover, we adapt them towards runtime-oriented losses, allowing for partially censored data while keeping a space- and time-complexity independent of the time horizon. In an extensive experimental evaluation on an adapted version of the ASlib benchmark, we demonstrate that theoretically well-founded methods based on Thompson sampling perform specifically strong and improve in comparison to existing methods.

Recently, the mainstream practice for training low-light raw image denoising methods has shifted towards employing synthetic data. Noise modeling, which focuses on characterizing the noise distribution of real-world sensors, profoundly influences the effectiveness and practicality of synthetic data. Currently, physics-based noise modeling struggles to characterize the entire real noise distribution, while learning-based noise modeling impractically depends on paired real data. In this paper, we propose a novel strategy: learning the noise model from dark frames instead of paired real data, to break down the data dependency. Based on this strategy, we introduce an efficient physics-guided noise neural proxy (PNNP) to approximate the real-world sensor noise model. Specifically, we integrate physical priors into neural proxies and introduce three efficient techniques: physics-guided noise decoupling (PND), physics-guided proxy model (PPM), and differentiable distribution loss (DDL). PND decouples the dark frame into different components and handles different levels of noise flexibly, which reduces the complexity of noise modeling. PPM incorporates physical priors to constrain the generated noise, which promotes the accuracy of noise modeling. DDL provides explicit and reliable supervision for noise distribution, which promotes the precision of noise modeling. PNNP exhibits powerful potential in characterizing the real noise distribution. Extensive experiments on public datasets demonstrate superior performance in practical low-light raw image denoising. The code will be available at https://github.com/fenghansen/PNNP.

We empirically study the effect of noise scheduling strategies for denoising diffusion generative models. There are three findings: (1) the noise scheduling is crucial for the performance, and the optimal one depends on the task (e.g., image sizes), (2) when increasing the image size, the optimal noise scheduling shifts towards a noisier one (due to increased redundancy in pixels), and (3) simply scaling the input data by a factor of b while keeping the noise schedule function fixed (equivalent to shifting the logSNR by log b) is a good strategy across image sizes. This simple recipe, when combined with recently proposed Recurrent Interface Network (RIN), yields state-of-the-art pixel-based diffusion models for high-resolution images on ImageNet, enabling single-stage, end-to-end generation of diverse and high-fidelity images at 1024times1024 resolution (without upsampling/cascades).

The Forward-Forward (FF) algorithm trains networks layer-by-layer using a local "goodness function," yet sum-of-squares (SoS) has remained the only choice studied. We systematically explore the goodness-function design space and identify a unifying principle: the goodness function must be sensitive to the shape of neural activity, not its total energy. This principle is motivated by the observation that deep network activations follow heavy-tailed distributions and that discriminative information is often concentrated in peak activities. We propose two complementary families: selective functions (top-k, entmax-weighted energy) that measure only peak activity, and shape-sensitive functions (excess kurtosis / "burstiness" and higher-order moments) that reward heavy-tailed distributions via scale-invariant statistics. Combined with separate label-feature forwarding (FFCL), controlled experiments across 13 goodness functions, 5 activations, 6 datasets, and three continuous sweeps, each tracing a characteristic inverted-U, yield 89.0% on Fashion-MNIST and 98.2+-0.1% on MNIST (4x2000), a +32.6pp gain over SoS, with consistent improvements across all benchmarks (+72pp USPS, +52pp SVHN). The scale-invariant nature of burstiness makes it particularly robust to magnitude shifts across layers and datasets.

Despite the remarkable success of diffusion models in image generation, slow sampling remains a persistent issue. To accelerate the sampling process, prior studies have reformulated diffusion sampling as an ODE/SDE and introduced higher-order numerical methods. However, these methods often produce divergence artifacts, especially with a low number of sampling steps, which limits the achievable acceleration. In this paper, we investigate the potential causes of these artifacts and suggest that the small stability regions of these methods could be the principal cause. To address this issue, we propose two novel techniques. The first technique involves the incorporation of Heavy Ball (HB) momentum, a well-known technique for improving optimization, into existing diffusion numerical methods to expand their stability regions. We also prove that the resulting methods have first-order convergence. The second technique, called Generalized Heavy Ball (GHVB), constructs a new high-order method that offers a variable trade-off between accuracy and artifact suppression. Experimental results show that our techniques are highly effective in reducing artifacts and improving image quality, surpassing state-of-the-art diffusion solvers on both pixel-based and latent-based diffusion models for low-step sampling. Our research provides novel insights into the design of numerical methods for future diffusion work.

We discover that common diffusion noise schedules do not enforce the last timestep to have zero signal-to-noise ratio (SNR), and some implementations of diffusion samplers do not start from the last timestep. Such designs are flawed and do not reflect the fact that the model is given pure Gaussian noise at inference, creating a discrepancy between training and inference. We show that the flawed design causes real problems in existing implementations. In Stable Diffusion, it severely limits the model to only generate images with medium brightness and prevents it from generating very bright and dark samples. We propose a few simple fixes: (1) rescale the noise schedule to enforce zero terminal SNR; (2) train the model with v prediction; (3) change the sampler to always start from the last timestep; (4) rescale classifier-free guidance to prevent over-exposure. These simple changes ensure the diffusion process is congruent between training and inference and allow the model to generate samples more faithful to the original data distribution.