Gaussian Noise

The Composed Image Retrieval (CIR) task provides a flexible retrieval paradigm via a reference image and modification text, but it heavily relies on expensive and error-prone triplet annotations. This paper systematically investigates the Noisy Triplet Correspondence (NTC) problem introduced by annotations. We find that NTC noise, particularly hard noise'' (i.e., the reference and target images are highly similar but the modification text is incorrect), poses a unique challenge to existing Noise Correspondence Learning (NCL) methods because it breaks the traditional small loss hypothesis''. We identify and elucidate three key, yet overlooked, challenges in the NTC task, namely (C1) Modality Suppression, (C2) Negative Anchor Deficiency, and (C3) Unlearning Backlash. To address these challenges, we propose a Cone-based robuSt noisE-unlearning comPositional network (ConeSep). Specifically, we first propose Geometric Fidelity Quantization, theoretically establishing and practically estimating a noise boundary to precisely locate noisy correspondence. Next, we introduce Negative Boundary Learning, which learns a ``diagonal negative combination'' for each query as its explicit semantic opposite-anchor in the embedding space. Finally, we design Boundary-based Targeted Unlearning, which models the noisy correction process as an optimal transport problem, elegantly avoiding Unlearning Backlash. Extensive experiments on benchmark datasets (FashionIQ and CIRR) demonstrate that ConeSep significantly outperforms current state-of-the-art methods, which fully demonstrates the effectiveness and robustness of our method.

Composed Image Retrieval (CIR) is a challenging image retrieval paradigm that enables to retrieve target images based on multimodal queries consisting of reference images and modification texts. Although substantial progress has been made in recent years, existing methods assume that all samples are correctly matched. However, in real-world scenarios, due to high triplet annotation costs, CIR datasets inevitably contain annotation errors, resulting in incorrectly matched triplets. To address this issue, the problem of Noisy Triplet Correspondence (NTC) has attracted growing attention. We argue that noise in CIR can be categorized into two types: cross-modal correspondence noise and modality-inherent noise. The former arises from mismatches across modalities, whereas the latter originates from intra-modal background interference or visual factors irrelevant to the coarse-grained modification annotations. However, modality-inherent noise is often overlooked, and research on cross-modal correspondence noise remains nascent. To tackle above issues, we propose the Invariance and discrimiNaTion-awarE Noise neTwork (INTENT), comprising two components: Visual Invariant Composition and Bi-Objective Discriminative Learning, specifically designed to handle the two-aspect noise. The former applies causal intervention on the visual side via Fast Fourier Transform (FFT) to generate intervened composed features, enforcing visual invariance and enabling the model to ignore modality-inherent noise during composition. The latter adopts collaborative optimization with both positive and negative samples, and constructs a scalable decision boundary that dynamically adjusts decisions based on the loyalty degree, enabling robust correspondence discrimination. Extensive experiments on two widely used benchmark datasets demonstrate the superiority and robustness of INTENT.

In multi-view clustering, the quality of different views may vary substantially, and low-quality or degraded views can impair overall clustering performance. However, existing studies mainly address this issue within the clustering process through view weighting or noise-robust optimization, while paying limited attention to data-level assessment before clustering. In this paper, we study the problem of pre-clustering noisy-view analysis in multi-view data from a clusterability perspective. To this end, we propose a Multi-View Clusterability Score (MVCS), which quantifies the strength of latent cluster-related structures in multi-view data through three complementary components: per-view structural clusterability, joint-space clusterability, and cross-view neighborhood consistency. To the best of our knowledge, this is the first clusterability score specifically designed for multi-view data. We further use it to perform potentially noisy view analysis and noisy-view detection before clustering. Extensive experiments on real-world datasets demonstrate that noisy views can significantly degrade clustering performance, and that, compared with existing clusterability measures designed for single-view data, the proposed method more effectively supports noisy-view analysis and detection.

Machine learning has become a powerful tool for discovering governing laws of dynamical systems from data. However, most existing approaches degrade severely when observations are sparse, noisy, or irregularly sampled. In this work, we address the problem of learning symbolic representations of nonlinear Hamiltonian dynamical systems under extreme data scarcity by explicitly incorporating physical structure into the learning architecture. We introduce Adaptable Symplectic Recurrent Neural Networks (ASRNNs), a parameter-cognizant, structure-preserving model that combines Hamiltonian learning with symplectic recurrent integration, avoiding time derivative estimation, and enabling stable learning under noise. We demonstrate that ASRNNs can accurately predict long-term dynamics even when each training trajectory consists of only two irregularly spaced time points, possibly corrupted by correlated noise. Leveraging ASRNNs as structure-preserving data generators, we further enable symbolic discovery using independent regression methods (SINDy and PySR), recovering exact symbolic equations for polynomial systems and consistent polynomial approximations for non-polynomial Hamiltonians. Our results show that such architectures can provide a robust pathway to interpretable discovery of Hamiltonian dynamics from sparse and noisy data.

We provide evidence that language models can detect, localize and, to a certain degree, verbalize the difference between perturbations applied to their activations. More precisely, we either (a) mask activations, simulating dropout, or (b) add Gaussian noise to them, at a target sentence. We then ask a multiple-choice question such as "Which of the previous sentences was perturbed?" or "Which of the two perturbations was applied?". We test models from the Llama, Olmo, and Qwen families, with sizes between 8B and 32B, all of which can easily detect and localize the perturbations, often with perfect accuracy. These models can also learn, when taught in context, to distinguish between dropout and Gaussian noise. Notably, Qwen3-32B's zero-shot accuracy in identifying which perturbation was applied improves as a function of the perturbation strength and, moreover, decreases if the in-context labels are flipped, suggesting a prior for the correct ones -- even modulo controls. Because dropout has been used as a training-regularization technique, while Gaussian noise is sometimes added during inference, we discuss the possibility of a data-agnostic "training awareness" signal and the implications for AI safety.

Cross-dose denoising for low-dose positron emission tomography (LDPET) has been proposed to address the limited generalization of models trained at a single noise level. However, neural networks trained on a specific dose level often fail to generalize to other dose conditions due to variations in noise magnitude and statistical properties. Conventional "one-size-for-all" models attempt to mitigate this variability but tend to learn averaged representations across noise levels, resulting in degraded performance. In this work, we analyze this limitation and show that standard training formulations implicitly optimize an expectation over heterogeneous noise distributions, causing the network to learn an averaged denoising mapping that cannot accurately model dose-specific noise characteristics. We propose a unified residual noise learning framework that estimates noise directly from low-dose PET images rather than predicting full-dose images. Experiments on large-scale multi-dose PET datasets from two medical centers demonstrate that the proposed method outperforms the "one-size-for-all" model, individual dose-specific U-Net models, and dose-conditioned approaches, achieving improved denoising performance. These results indicate that residual noise learning effectively mitigates the averaging effect and enhances generalization for cross-dose PET denoising.

Diffusion Probabilistic Models have demonstrated remarkable performance across a wide range of generative tasks. However, we have observed that these models often suffer from a Signal-to-Noise Ratio-timestep (SNR-t) bias. This bias refers to the misalignment between the SNR of the denoising sample and its corresponding timestep during the inference phase. Specifically, during training, the SNR of a sample is strictly coupled with its timestep. However, this correspondence is disrupted during inference, leading to error accumulation and impairing the generation quality. We provide comprehensive empirical evidence and theoretical analysis to substantiate this phenomenon and propose a simple yet effective differential correction method to mitigate the SNR-t bias. Recognizing that diffusion models typically reconstruct low-frequency components before focusing on high-frequency details during the reverse denoising process, we decompose samples into various frequency components and apply differential correction to each component individually. Extensive experiments show that our approach significantly improves the generation quality of various diffusion models (IDDPM, ADM, DDIM, A-DPM, EA-DPM, EDM, PFGM++, and FLUX) on datasets of various resolutions with negligible computational overhead. The code is at https://github.com/AMAP-ML/DCW.

M-MIMO is one of the crucial technologies for increasing spectral and energy efficiency of wireless networks. Most of the current works assume that M-MIMO arrays are equipped with a linear front end. However, ongoing efforts to make wireless networks more energy-efficient push the hardware to the limits, where its nonlinear behavior appears. This is especially a common problem for the multicarrier systems, e.g., OFDM used in 4G, 5G, and possibly also in 6G, which is characterized by a high Peak-to-Average Power Ratio. While the impact of a nonlinear Power Amplifier (PA) on an OFDM signal is well characterized, it is a relatively new topic for the M-MIMO OFDM systems. Most of the recent works either neglect nonlinear effects or utilize simplified models proper for Rayleigh or LoS radio channel models. In this paper, we first theoretically characterize the nonlinear distortion in the M-MIMO system under commonly used radio channel models. Then, utilizing 3D-Ray Tracing (3D-RT) software, we demonstrate that these models are not very accurate. Instead, we propose two models: a statistical one and an ML-based one using 3D-RT results. The proposed statistical model utilizes the Generalized Extreme Value (GEV) distribution to model Signal to Distortion Ratio (SDR) for victim users, receiving nonlinear distortion, e.g., as interference from neighboring cells. The proposed ML model aims to predict SDR for a scheduled user (receiving nonlinear distortion along with the desired signal), based on the spatial characteristics of the radio channel and the operation point of each PA feeding at the M-MIMO antenna array. The predicted SDR can then be used to perform PA-aware per-user power allocation. The results show about 12% median gain in user throughput achieved by the proposed ML-based power allocation scheme over the state-of-the-art, fixed operating point scheme.

Medical image denoising (MID) lacks absolutely clean images for supervision, leading to a noisy reference problem that fundamentally limits denoising performance. Existing simulated-supervised discriminative learning (SimSDL) and simulated-supervised generative learning (SimSGL) treat noisy references as clean targets, causing suboptimal convergence or reference-biased learning, while self-supervised learning (SSL) imposes restrictive noise assumptions that are seldom satisfied in realistic MID scenarios. We propose \textbf{RelativeFlow}, a flow matching framework that learns from heterogeneous noisy references and drives inputs from arbitrary quality levels toward a unified high-quality target. RelativeFlow reformulates flow matching by decomposing the absolute noise-to-clean mapping into relative noisier-to-noisy mappings, and realizes this formulation through two key components: 1) consistent transport (CoT), a displacement map that constrains relative flows to be components of and progressively compose a unified absolute flow, and 2) simulation-based velocity field (SVF), which constructs a learnable velocity field using modality-specific degradation operators to support different medical imaging modalities. Extensive experiments on Computed Tomography (CT) and Magnetic Resonance (MR) denoising demonstrate that RelativeFlow significantly outperforms existing methods, taming MID with noisy references.

Deepfake detectors face growing challenges in generalization as new image synthesis techniques emerge. In particular, deepfakes generated by diffusion models are highly photorealistic and often evade detectors trained on GAN-based forgeries. This paper addresses the generalization problem in deepfake detection by leveraging diffusion noise characteristics. We propose an Attention-guided Noise Learning (ANL) framework that integrates a pre-trained diffusion model into the deepfake detection pipeline to guide the learning of more robust features. Specifically, our method uses the diffusion model's denoising process to expose subtle artifacts: the detector is trained to predict the noise contained in an input image at a given diffusion step, forcing it to capture discrepancies between real and synthetic images, while an attention-guided mechanism derived from the predicted noise is introduced to encourage the model to focus on globally distributed discrepancies rather than local patterns. By harnessing the frozen diffusion model's learned distribution of natural images, the ANL method acts as a form of regularization, improving the detector's generalization to unseen forgery types. Extensive experiments demonstrate that ANL significantly outperforms existing methods on multiple benchmarks, achieving state-of-the-art accuracy in detecting diffusion-generated deepfakes. Notably, the proposed framework boosts generalization performance (e.g., improving ACC/AP by a substantial margin on unseen models) without introducing additional overhead during inference. Our results highlight that diffusion noise provides a powerful signal for generalizable deepfake detection.

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Our analysis considers trajectory-dependent regressors and allows marginally stable dynamics with polynomial mean-square state growth. We prove that the diameter of the feasible parameter set shrinks with sample complexity $\widetilde{\mathcal O}(1/ε)$ where $ε$ is the estimation error. Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.

Long-form speech recognition with large encoder-decoder models such as Whisper often exhibit hallucinations, repetition loops, and content omissions. These errors can accumulate and be further amplified when the previous segment's transcription is used as decoding context. We propose Whisper-CD, a training-free contrastive decoding framework that contrasts clean-audio logits against negative logits computed from three acoustically motivated perturbations: Gaussian noise injection, silence signal, and audio temporal shift. We aggregate these negatives via the log-sum-exp operator, building a unified multi-negative objective for token-by-token decoding. Across five English long-form benchmarks, Whisper-CD reduces WER by up to 24.3pp on CORAAL and shows 48% faster token generation throughput than beam search. Because Whisper-CD operates purely at inference time, it can be applied as a drop-in replacement to already-deployed Whisper systems without retraining.

Diffusion distillation is central to accelerating image and video generation, yet existing methods are fundamentally limited by the denoising process, where step reduction has largely saturated. Partial timestep low-resolution generation can further accelerate inference, but it suffers noticeable quality degradation due to cross-resolution distribution gaps. We propose Cross-Resolution Distribution Matching Distillation (RMD), a novel distillation framework that bridges cross-resolution distribution gaps for high-fidelity, few-step multi-resolution cascaded inference. Specifically, RMD divides the timestep intervals for each resolution using logarithmic signal-to-noise ratio (logSNR) curves, and introduces logSNR-based mapping to compensate for resolution-induced shifts. Distribution matching is conducted along resolution trajectories to reduce the gap between low-resolution generator distributions and the teacher's high-resolution distribution. In addition, a predicted-noise re-injection mechanism is incorporated during upsampling to stabilize training and improve synthesis quality. Quantitative and qualitative results show that RMD preserves high-fidelity generation while accelerating inference across various backbones. Notably, RMD achieves up to 33.4X speedup on SDXL and 25.6X on Wan2.1-14B, while preserving high visual fidelity.

Large audio language models (LALMs) are a class of foundation models for audio understanding. Existing LALMs tend to degrade significantly in real-world noisy acoustic conditions where speech and non-speech sounds interfere. While noise-aware fine-tuning can improve robustness, it requires task-specific noisy data and expensive retraining, limiting scalability. To address this issue, we propose Focus-Then-Listen (FTL), a plug-and-play audio enhancer that improves LALMs' noise robustness. Specifically, FTL first separates the input waveform into speech and non-speech, and a modality router is applied to predict the target audio modality (e.g., speech) based on the user's instruction. Finally, a modality-aware fusion block generates a task-adaptive enhanced signal for improved downstream perception and reasoning. Experiments across multiple LALMs and tasks show that FTL improves performance across different noise levels without fine-tuning on LALMs.

Deep multi-view clustering has achieved remarkable progress but remains vulnerable to complex noise in real-world applications. Existing noisy robust methods predominantly rely on a simplified binary assumption, treating data as either perfectly clean or completely corrupted. This overlooks the prevalent existence of heterogeneous observation noise, where contamination intensity varies continuously across data. To bridge this gap, we propose a novel framework termed Quality-Aware Robust Multi-View Clustering (QARMVC). Specifically, QARMVC employs an information bottleneck mechanism to extract intrinsic semantics for view reconstruction. Leveraging the insight that noise disrupts semantic integrity and impedes reconstruction, we utilize the resulting reconstruction discrepancy to precisely quantify fine-grained contamination intensity and derive instance-level quality scores. These scores are integrated into a hierarchical learning strategy: at the feature level, a quality-weighted contrastive objective is designed to adaptively suppress the propagation of noise; at the fusion level, a high-quality global consensus is constructed via quality-weighted aggregation, which is subsequently utilized to align and rectify local views via mutual information maximization. Extensive experiments on five benchmark datasets demonstrate that QARMVC consistently outperforms state-of-the-art baselines, particularly in scenarios with heterogeneous noise intensities.

Unsupervised remote sensing change detection (UCD) aims to localise changes between two images of the same region without relying on labelled training data. Most recent approaches either use a frozen foundation model in a training-free manner or train with synthetic changes generated in pixel space. Both strategies inherently rely on predefined assumptions about change types, typically introduced through handcrafted rules, external datasets, or auxiliary generative models. Due to these assumptions, such methods fail to generalise beyond a few change types, limiting their real-world usage, especially in rare or complex scenarios. To address this, we propose MaSoN (Make Some Noise), an end-to-end UCD framework that synthesises diverse changes directly in the latent feature space during training. It generates changes dynamically estimated from feature statistics of the target data, enabling diverse yet data-driven variation aligned with the target domain. Since synthesis happens in latent space, it also easily extends to new modalities, such as SAR and multispectral data. MaSoN generalises strongly across diverse change types and improves the average F1 score across five benchmarks by 14.1 percentage points. Project page: https://blaz-r.github.io/mason_ucd/

Spoken query retrieval is an important interaction mode in modern information retrieval. However, existing evaluation datasets are often limited to simple queries under constrained noise conditions, making them inadequate for assessing the robustness of spoken query retrieval systems under complex acoustic perturbations. To address this limitation, we present SQuTR, a robustness benchmark for spoken query retrieval that includes a large-scale dataset and a unified evaluation protocol. SQuTR aggregates 37,317 unique queries from six commonly used English and Chinese text retrieval datasets, spanning multiple domains and diverse query types. We synthesize speech using voice profiles from 200 real speakers and mix 17 categories of real-world environmental noise under controlled SNR levels, enabling reproducible robustness evaluation from quiet to highly noisy conditions. Under the unified protocol, we conduct large-scale evaluations on representative cascaded and end-to-end retrieval systems. Experimental results show that retrieval performance decreases as noise increases, with substantially different drops across systems. Even large-scale retrieval models struggle under extreme noise, indicating that robustness remains a critical bottleneck. Overall, SQuTR provides a reproducible testbed for benchmarking and diagnostic analysis, and facilitates future research on robustness in spoken query to text retrieval.

Multivariate data often exhibit complex dependencies that violate the assumption of isotropic residual noise. For such cases, we introduce Graph Regularized PCA (GR-PCA). It is a graph-based regularization of PCA that incorporates the dependency structure of the data features by learning a sparse precision graph and biasing loadings toward the low-frequency Fourier modes of the corresponding graph Laplacian. Consequently, high-frequency signals are suppressed, while graph-coherent low-frequency ones are preserved, yielding interpretable principal components aligned with conditional relationships. We evaluate GR-PCA on synthetic data spanning diverse graph topologies, signal-to-noise ratios, and sparsity levels. Compared to mainstream alternatives, it concentrates variance on the intended support, produces loadings with lower graph-Laplacian energy, and remains competitive in out-of-sample reconstruction. When high-frequency signals are present, the graph Laplacian penalty prevents overfitting, reducing the reconstruction accuracy but improving structural fidelity. The advantage over PCA is most pronounced when high-frequency signals are graph-correlated, whereas PCA remains competitive when such signals are nearly rotationally invariant. The procedure is simple to implement, modular with respect to the precision estimator, and scalable, providing a practical route to structure-aware dimensionality reduction that improves structural fidelity without sacrificing predictive performance.

As a critical application of computational intelligence in remote sensing, deep learning-based synthetic aperture radar (SAR) image target recognition facilitates intelligent perception but typically relies on centralized training, where multi-source SAR data are uploaded to a single server, raising privacy and security concerns. Federated learning (FL) provides an emerging computational intelligence paradigm for SAR image target recognition, enabling cross-site collaboration while preserving local data privacy. However, FL confronts critical security risks, where malicious clients can exploit SAR's multiplicative speckle noise to conceal backdoor triggers, severely challenging the robustness of the computational intelligence model. To address this challenge, we propose NADAFD, a noise-aware and dynamically adaptive federated defense framework that integrates frequency-domain, spatial-domain, and client-behavior analyses to counter SAR-specific backdoor threats. Specifically, we introduce a frequency-domain collaborative inversion mechanism to expose cross-client spectral inconsistencies indicative of hidden backdoor triggers. We further design a noise-aware adversarial training strategy that embeds $Γ$-distributed speckle characteristics into mask-guided adversarial sample generation to enhance robustness against both backdoor attacks and SAR speckle noise. In addition, we present a dynamic health assessment module that tracks client update behaviors across training rounds and adaptively adjusts aggregation weights to mitigate evolving malicious contributions. Experiments on MSTAR and OpenSARShip datasets demonstrate that NADAFD achieves higher accuracy on clean test samples and a lower backdoor attack success rate on triggered inputs than existing federated backdoor defenses for SAR target recognition.

Randomized Smoothing (RS) is a prominent technique for certifying the robustness of neural networks against adversarial perturbations. With RS, achieving high accuracy at small radii requires a small noise variance, while achieving high accuracy at large radii requires a large noise variance. However, the global noise variance used in the standard RS formulation leads to a fundamental limitation: there exists no global noise variance that simultaneously achieves strong performance at both small and large radii. To break through the global variance limitation, we propose a dual RS framework which enables input-dependent noise variances. To achieve that, we first prove that RS remains valid with input-dependent noise variances, provided the variance is locally constant around each input. Building on this result, we introduce two components: (i) a variance estimator predicts an optimal noise variance for each input, (ii) this estimated variance is then used by a standard RS classifier. The variance estimator is independently smoothed via RS to ensure local constancy, enabling flexible design. We also introduce training strategies to iteratively optimize the two components. Experiments on CIFAR-10 demonstrate that our dual RS method provides strong performance for both small and large radii-unattainable with global noise variance-while incurring only a 60% computational overhead at inference. Moreover, it outperforms prior input-dependent noise approaches across most radii, with gains at radii 0.5, 0.75, and 1.0 of 15.6%, 20.0%, and 15.7%. On ImageNet, dual RS remains effective across all radii, with advantages of 8.6%, 17.1%, and 9.1% at radii 0.5, 1.0, and 1.5. Additionally, the dual RS framework provides a routing perspective for certified robustness, improving the accuracy-robustness trade-off with off-the-shelf expert RS models.

To distinguish Markov equivalent graphs in causal discovery, it is necessary to restrict the structural causal model. Crucially, we need to be able to distinguish cause $X$ from effect $Y$ in bivariate models, that is, distinguish the two graphs $X \to Y$ and $Y \to X$. Location-scale noise models (LSNMs), in which the effect $Y$ is modeled based on the cause $X$ as $Y = f(X) + g(X)N$, form a flexible class of models that is general and identifiable in most cases. Estimating these models for arbitrary noise terms $N$, however, is challenging. Therefore, practical estimators are typically restricted to symmetric distributions, such as the normal distribution. As we showcase in this paper, when $N$ is a skewed random variable, which is likely in real-world domains, the reliability of these approaches decreases. To approach this limitation, we propose SkewD, a likelihood-based algorithm for bivariate causal discovery under LSNMs with skewed noise distributions. SkewD extends the usual normal-distribution framework to the skew-normal setting, enabling reliable inference under symmetric and skewed noise. For parameter estimation, we employ a combination of a heuristic search and an expectation conditional maximization algorithm. We evaluate SkewD on novel synthetically generated datasets with skewed noise as well as established benchmark datasets. Throughout our experiments, SkewD exhibits a strong performance and, in comparison to prior work, remains robust under high skewness.

In high-stakes scenarios, such as medical imaging applications, it is critical to equip the predictions of a regression model with reliable confidence intervals. Recently, Conformal Prediction (CP) has emerged as a powerful statistical framework that, based on a labeled calibration set, generates intervals that include the true labels with a pre-specified probability. In this paper, we address the problem of applying CP for regression models when the calibration set contains noisy labels. We begin by establishing a mathematically grounded procedure for estimating the noise-free CP threshold. Then, we turn it into a practical algorithm that overcomes the challenges arising from the continuous nature of the regression problem. We evaluate the proposed method on two medical imaging regression datasets with Gaussian label noise. Our method significantly outperforms the existing alternative, achieving performance close to the clean-label setting.

We consider the problem of support recovery for sparse binary signals from noisy linear measurements. For sparse Gaussian measurement matrices we identify sufficient conditions on the minimal sample size for maximum-likelihood recovery in the high-SNR regime $ds/p \to \infty$, where $p$ denotes the signal dimension, $s$ the number of non-zero components of the signal, and $d$ the expected number of non-zero components per row of measurement. Combined with known lower bounds, this yields an information-theoretic threshold of order $s\log(p/s) / \log(ds/p)$, making explicit the price of measurement sparsity. In particular, we highlight a regime where the sample-complexity loss from measurement sparsity is logarithmic while the computational gain is nearly linear. Second, we study recovery after sparsifying an originally dense Gaussian design: the observations are generated from the dense design, while estimation uses an independently sparsified design and a rescaled response. In the proportional regime $s=αp$, $d=ψp$, we prove that, for every fixed target error level $δ$ and every slack $\varepsilon>0$, a sample size of order $p/ψ^2$ is sufficient for support recovery for arbitrarily small $ψ$.

Objective: Sparse Bayesian learning provides an effective framework to solve high-dimensional problems in brain signal decoding. However, conventional likelihoods regarding data distributions, such as Gaussian or Bernoulli, are potentially inadequate for handling the noisy recordings of brain activity. Hence, this work aims to formulate a robust sparse Bayesian learning framework to address noisy high-dimensional brain activity decoding. Methods: Motivated by the commendable robustness of the minimum error entropy learning criterion for addressing non-Gaussian signals, this study reformulated the sparse Bayesian learning framework under a generalized Bayesian paradigm, in which the model parameter is regulated with the minimum error entropy loss rather than a conventional likelihood function. Results: Our developed SBL-MEE algorithm was evaluated with two real-world brain decoding tasks of regression and classification scenarios, respectively. Experimental results demonstrated that our approach not only realizes superior brain decoding performance than existing methods, but also presents more physiologically interpretable decoder patterns. Conclusion: Although minimum error entropy is not constructed from an arbitrary probabilistic distribution, it is effective to establish noise-robust inference in sparse Bayesian learning method. Significance: This work provides a powerful tool to improve brain activity decoding capability, particularly regarding the noisy high-dimensional setting, thus promoting biomedical engineering applications such as brain-computer interface.

Model-free and reinforcement learning-based adaptive filtering methods are gaining traction for denoising in dynamic, non-stationary environments such as wireless signal channels, biomedical monitoring, and sensor networks. Traditional filters such as LMS, RLS, Wiener, and Kalman are often limited by assumptions of stationarity, the need for exact noise statistics, or fragile parameter tuning. This paper proposes an adaptive filtering framework using Proximal Policy Optimization (PPO), guided by a composite reward that balances SNR improvement, MSE reduction, and residual smoothness. We frame adaptive filtering as a Markov decision process and train a PPO agent to adjust filter coefficients directly in response to changing noise. Experiments on synthetic nonstationary signals with diverse noise types show that the PPO agent generalizes beyond its training distribution. Moreover, real-world analysis is made and evaluated on ECG recordings from the MIT-BIH Noise Stress Test Database corrupted by baseline wander, electrode motion, and muscle artifacts. The learned PPO policy achieves real-time inference and slightly outperforms strong classical baselines on ECG denoising. These results demonstrate the viability of policy-gradient reinforcement learning as a computationally efficient and flexible tool for adaptive filtering in nonlinear, time-varying dynamical systems.

The practical success of deep learning has led to the discovery of several surprising phenomena. One of these phenomena, that has spurred intense theoretical research, is ``benign overfitting'': deep neural networks seem to generalize well in the over-parametrized regime even though the networks show a perfect fit to noisy training data. It is now known that benign overfitting also occurs in various classical statistical models. For linear maximum margin classifiers, benign overfitting has been established theoretically in a class of mixture models with very strong assumptions on the covariate distribution. However, even in this simple setting, many questions remain open. For instance, most of the existing literature focuses on the noiseless case where all true class labels are observed without errors, whereas the more interesting noisy case remains poorly understood. We provide a comprehensive study of benign overfitting for linear maximum margin classifiers. We discover a phase transition in test error bounds for the noisy model which was previously unknown and provide some geometric intuition behind it. We further considerably relax the required covariate assumptions in both the noisy and noiseless cases. Our results demonstrate that benign overfitting of maximum margin classifiers holds in a much wider range of scenarios than was previously known and provide new insights into the underlying mechanisms.

We consider the effect of Gaussian perturbations on least-squares residuals, orthogonal projections, and QR-type algorithms. The problem that motivated our investigations is as follows: suppose that a full column-rank matrix $B\in\mathbb{R}^{m\times n}$ has already been computed, and suppose that a new normalized column $q=(x+y)/\|x+y\|_2$ is to be appended to $B$, where $x\perp\operatorname{span}(B)$ is the ideal orthogonal component and $y$ represents the orthogonalization error. How large can the condition number $κ([B,q])$ of the resulting matrix $[B,q]$ become? While we provide a Weyl-type bound on the singular values of $[B,q]$, in terms of the extremal singular values of $B$ and the quantity $\|B^T y\|_2/\|x+y\|_2$, we also derive exact probability laws for norms and projection residuals under Gaussian perturbations. Finally, we use these probability laws to derive probabilistic condition-number bounds for QR-type processes with imperfect orthogonalization and exact normalization.

Perturbation robustness evaluates the vulnerabilities of models, arising from a variety of perturbations, such as data corruptions and adversarial attacks. Understanding the mechanisms of perturbation robustness is critical for global interpretability. We present a model-agnostic, global mechanistic interpretability method to interpret the perturbation robustness of image models. This research is motivated by two key aspects. First, previous global interpretability works, in tandem with robustness benchmarks, e.g. mean corruption error (mCE), are not designed to directly interpret the mechanisms of perturbation robustness within image models. Second, we notice that the spectral signal-to-noise ratios (SNR) of perturbed natural images exponentially decay over the frequency. This power-law-like decay implies that: Low-frequency signals are generally more robust than high-frequency signals -- yet high classification accuracy can not be achieved by low-frequency signals alone. By applying Shapley value theory, our method axiomatically quantifies the predictive powers of robust features and non-robust features within an information theory framework. Our method, dubbed as \textbf{I-ASIDE} (\textbf{I}mage \textbf{A}xiomatic \textbf{S}pectral \textbf{I}mportance \textbf{D}ecomposition \textbf{E}xplanation), provides a unique insight into model robustness mechanisms. We conduct extensive experiments over a variety of vision models pre-trained on ImageNet to show that \textbf{I-ASIDE} can not only \textbf{measure} the perturbation robustness but also \textbf{provide interpretations} of its mechanisms.

In this article, we introduce a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.