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Theory · Reinforcement Learning and Planning

Anand Brahmbhatt, Gon Buzaglo, Sofiia Druchyna, Elad Hazan

We propose a new method for the problem of controlling linear dynamical systems under partial observation and adversarial disturbances. Our new algorithm, Double Spectral Control (DSC), matches the best known regret guarantees while exponentially improving runtime complexity over previous approaches in its dependence on the system's stability margin. Our key innovation is a two-level spectral approximation strategy, leveraging double convolution with a universal basis of spectral filters, enabling efficient and accurate learning of the best linear dynamical controllers.

Computer Vision · Everything Else

Xingxing Yang, Jie Chen, Zaifeng Yang

Spectral reconstruction from RGB images often suffers from a metameric dilemma, where distinct spectral distributions map to nearly identical RGB values, making them indistinguishable to current models and leading to unreliable reconstructions. In this paper, we present Diff-Spectra that integrates supervised physics-aware spectral estimation and unsupervised high-fidelity spectral regularization for HSI reconstruction. We first introduce an Adaptive illumiChroma Decoupling (AICD) module to decouple illumination and chrominance information, which learns intrinsic and distinctive feature distributions, thereby mitigating the metameric issue. Then, we incorporate the AICD into a learnable spectral response function (SRF) guided hyperspectral initial estimation mechanism to mimic the physical image formation and thus inject physics-aware reasoning into neural networks, turning an ill-posed problem into a constrained, interpretable task. We also introduce a metameric spectra augmentation method to synthesize comprehensive hyperspectral data to pre-train a Spectral Diffusion Module (SDM), which internalizes the statistical properties of real-world HSI data, enforcing unsupervised high-fidelity regularization on the spectral transitions via inner-loop optimization during inference. Extensive experimental evaluations demonstrate that our Diff-Spectra achieves SOTA performance on both Spectral reconstruction and downstream HSI classification.

Theory · Learning Theory

Yuxin Ma, Dmitriy Kunisky

We introduce a new family of algorithms for detecting and estimating a rank-one signal from a noisy observation under prior information about that signal's direction, focusing on examples where the signal is known to have entries biased to be positive. Given a matrix observation $\mathbf{Y}$, our algorithms construct a *nonlinear Laplacian*, another matrix of the form $\mathbf{Y} + \mathrm{diag}(\sigma(\mathbf{Y1}))$ for a nonlinear $\sigma: \mathbb{R} \to \mathbb{R}$, and examine the top eigenvalue and eigenvector of this matrix. When $\mathbf{Y}$ is the (suitably normalized) adjacency matrix of a graph, our approach gives a class of algorithms that search for unusually dense subgraphs by computing a spectrum of the graph "deformed" by the degree profile $\mathbf{Y1}$. We study the performance of such algorithms compared to direct spectral algorithms (the case $\sigma = 0$) on models of sparse principal component analysis with biased signals, including the Gaussian planted submatrix problem. For such models, we rigorously characterize the strength of rank-one signal, as a function of the nonlinearity $\sigma$, required for an outlier eigenvalue to appear in the spectrum of a nonlinear Laplacian matrix. While identifying the $\sigma$ that minimizes the required signal strength in closed form seems intractable, we explore three approaches to design $\sigma$ numerically: exhaustively searching over simple classes of $\sigma$, learning $\sigma$ from datasets of problem instances, and tuning $\sigma$ using black-box optimization of the critical signal strength. We find both theoretically and empirically that, if $\sigma$ is chosen appropriately, then nonlinear Laplacian spectral algorithms substantially outperform direct spectral algorithms, while retaining the conceptual simplicity of spectral methods compared to broader classes of computations like approximate message passing or general first order methods.

Deep Learning · Everything Else

Jiawei Gu, Ziyue Qiao, Xinming Li, Zechao Li

Global Covariance Pooling (GCP) has garnered increasing attention in visual recognition tasks, where second-order statistics frequently yield stronger representations than first-order approaches. However, two main streams of GCP---Newton--Schulz-based iSQRT-COV and exact or near-exact SVD methods---struggle at opposite ends of the training spectrum. While iSQRT-COV stabilizes early learning by avoiding large gradient explosions, it over-compresses significant eigenvalues in later stages, causing an \emph{over-flattening} phenomenon that stalls final accuracy. In contrast, SVD-based methods excel at preserving the high-eigenvalue structure essential for deep networks but suffer from sensitivity to small eigenvalue gaps early on. We propose \textbf{Halley-SVD}, a high-order iterative method that unites the smooth gradient advantages of iSQRT-COV with the late-stage fidelity of SVD. Grounded in Halley's iteration, our approach obviates explicit divisions by $(\lambda_i - \lambda_j)$ and forgoes threshold- or polynomial-based heuristics. As a result, it prevents both early gradient explosions and the excessive compression of large eigenvalues. Extensive experiments on CNNs and transformer architectures show that Halley-SVD consistently and robustly outperforms iSQRT-COV at large model scales and batch sizes, achieving higher overall accuracy without mid-training switches or custom truncations. This work provides a new solution to the long-standing dichotomy in GCP, illustrating how high-order methods can balance robustness and spectral precision to fully harness the representational power of modern deep networks.

Theory · Domain Adaptation and Transfer Learning

Konstantinos Dafnis, Dimitris Metaxas

Vision–language models (VLMs) excel at zero-shot inference but often degrade under test-time domain shifts. For this reason, episodic test-time adaptation strategies have recently emerged as powerful techniques for adapting VLMs to a single unlabeled image. However, existing adaptation strategies, such as test-time prompt tuning, typically require backpropagating through large encoder weights or altering core model components. In this work, we introduce \textbf{S}pectrum-Aware \textbf{T}est-Time \textbf{S}teering (\textbf{STS}), a \textit{lightweight adaptation framework} that extracts a spectral subspace from the textual embeddings to define principal semantic directions, and learns to steer latent representations in a spectrum-aware manner by adapting a small number of per-sample shift parameters to minimize entropy across augmented views. STS operates entirely at inference in the latent space, without backpropagation through or modification of the frozen encoders. Building on standard evaluation protocols, our comprehensive experiments demonstrate that STS largely surpasses or compares favorably against state-of-the-art test-time adaptation methods, while introducing only a handful of additional parameters and achieving inference speeds up to 8× faster with a 12× smaller memory footprint than conventional test-time prompt tuning. The code is available at \url{https://github.com/kdafnis/STS}.

Deep Learning · Graph Neural Networks

Yi Li, Zhichun Guo, Guanpeng Li, Bingzhe Li

Training Graph Neural Networks (GNNs) often relies on repeated, irregular, and expensive message-passing operations over all nodes (e.g., $N$), leading to high computational overhead. To alleviate this inefficiency, we revisit the GNNs training from a spectral perspective. In many real-world graphs, node features and embeddings exhibit sparse representation in the Graph Fourier domain. This inherent spectral sparsity aligns well with the principles of Compressed Sensing, which posits that signals sparse in one transform domain can be accurately reconstructed from a significantly reduced number of measurements. This observation motivates the design of a more efficient GNNs that operates predominantly in compressed spectral subspace. Thus, we propose You Only Spectralize Once (YOSO), a GNN training scheme that performs single Graph Fourier Transformation to project features onto a learnable orthonormal Fourier basis, retaining only $M$ spectral coefficients ($M \ll N$). The entire GNN computation is then carried out in reduced spectral domain. Final full-graph embeddings are recovered only at output layer by solving a bounded $\ell_{2,1}$-regularized optimization problem. Theoretically, drawing upon Compressed Sensing theory, we prove stable recovery throughout training by showing that the projection onto our learnable Fourier basis can satisfy the Restricted Isometry Property when $M=\mathcal{O}(k \log N)$ for $k$-row-sparse spectra, acting as the measurement process. Empirically, YOSO achieves an average 74\% reduction in training time across five benchmark datasets compared to state-of-the-art methods, while maintaining competitive accuracy.

Deep Learning · Graph Neural Networks

Katharina Limbeck, Lydia Mezrag, Guy Wolf, Bastian Rieck

Graph Neural Networks (GNNs) have shown significant success for graph-based tasks. Motivated by the prevalence of large datasets in real-world applications, pooling layers are crucial components of GNNs. By reducing the size of input graphs, pooling enables faster training and potentially better generalisation. However, existing pooling operations often optimise for the learning task at the expense of discarding fundamental graph structures, thus reducing interpretability. This leads to unreliable performance across dataset types, downstream tasks and pooling ratios. Addressing these concerns, we propose novel graph pooling layers for structure-aware pooling via edge collapses. Our methods leverage diffusion geometry and iteratively reduce a graph’s size while preserving both its metric structure and its structural diversity. We guide pooling using *magnitude*, an isometry-invariant diversity measure, which permits us to control the fidelity of the pooling process. Further, we use the *spread* of a metric space as a faster and more stable alternative ensuring computational efficiency. Empirical results demonstrate that our methods (i) achieve top performance compared to alternative pooling layers across a range of diverse graph classification tasks, (ii) preserve key spectral properties of the input graphs, and (iii) retain high accuracy across varying pooling ratios.

Probabilistic Methods · Monte Carlo and Sampling Methods

Jhanvi Garg, Krishnakumar Balasubramanian, Quan Zhou

Simulated tempering is a widely used strategy for sampling from multimodal distributions. In this paper, we consider simulated tempering combined with an arbitrary local Markov chain Monte Carlo sampler and present a new decomposition theorem that provides a lower bound on the restricted spectral gap of the algorithm for sampling from mixture distributions. By working with the restricted spectral gap, the applicability of our results is extended to broader settings such as when the usual spectral gap is difficult to bound or becomes degenerate. We demonstrate the application of our theoretical results by analyzing simulated tempering combined with random walk Metropolis--Hastings for sampling from mixtures of Gaussian distributions. Our complexity bound scales polynomially with the separation between modes, logarithmically with $1/\varepsilon$, where $\varepsilon$ denotes the target accuracy in total variation distance, and exponentially with the dimension $d$.

General Machine Learning · Everything Else

Ido Cohen, Ronen Talmon

We propose a novel functorial graph coarsening method that preserves inner products between node features, a property often overlooked by existing approaches focusing primarily on structural fidelity. By treating node features as functions on the graph and preserving their inner products, our method retains both structural and feature relationships, facilitating substantial benefits for downstream tasks. To formalize this, we introduce the Inner Product Error (IPE), which quantifies how the inner products between node features are preserved. Leveraging the underlying geometry of the problem on the Grassmann manifold, we formulate an optimization objective that minimizes the IPE, also for unseen smooth functions. We show that minimizing the IPE improves standard coarsening metrics, and illustrate our method’s properties through visual examples that highlight its clustering ability. Empirical results on benchmarks for graph coarsening and node classification show that our approach outperforms existing state-of-the-art methods.

Theory · Learning Theory

Chen Fan, Mark Schmidt, Christos Thrampoulidis

Different gradient-based methods for optimizing overparameterized models can all achieve zero training error yet converge to distinctly different solutions inducing different generalization properties. We provide the first complete characterization of implicit optimization bias for p-norm normalized steepest descent (NSD) and momentum steepest descent (NMD) algorithms in multi-class linear classification with cross-entropy loss. Our key theoretical contribution is proving that these algorithms converge to solutions maximizing the margin with respect to the classifier matrix's p-norm, with established convergence rates. These results encompass important special cases including Spectral Descent and Muon, which we show converge to max-margin solutions with respect to the spectral norm. A key insight of our contribution is that the analysis of general entry-wise and Schatten p-norms can be reduced to the analysis of NSD/NMD with max-norm by exploiting a natural ordering property between all p-norms relative to the max-norm and its dual sum-norm. For the specific case of descent with respect to the max-norm, we further extend our analysis to include preconditioning, showing that Adam converges to the matrix's max-norm solution. Our results demonstrate that the multi-class linear setting, which is inherently richer than the binary counterpart, provides the most transparent framework for studying implicit biases of matrix-parameter optimization algorithms.

Theory · Learning Theory

Leonardo Defilippis, Yatin Dandi, Pierre Mergny, Florent Krzakala, Bruno Loureiro

We consider the problem of how many samples from a Gaussian multi-index model are required to weakly reconstruct the relevant index subspace. Despite its increasing popularity as a testbed for investigating the computational complexity of neural networks, results beyond the single-index setting remain elusive. In this work, we introduce spectral algorithms based on the linearization of a message passing scheme tailored to this problem. Our main contribution is to show that the proposed methods achieve the optimal reconstruction threshold. Leveraging a high-dimensional characterization of the algorithms, we show that above the critical threshold the leading eigenvector correlates with the relevant index subspace, a phenomenon reminiscent of the Baik–Ben Arous–Peche (BBP) transition in spiked models arising in random matrix theory. Supported by numerical experiments and a rigorous theoretical framework, our work bridges critical gaps in the computational limits of weak learnability in multi-index model.

Deep Learning · Theory

Hemanth Saratchandran, Simon Lucey

We present a theoretical analysis of the Jacobian of a attention block within a transformer, showing that it is governed by the query, key, and value projections that define the attention mechanism. Leveraging this insight, we introduce a method that systematically alters the spectral properties of each attention layer to reduce the Jacobian’s condition number, thereby improving the overall conditioning of the attention layers within a transformer network. We empirically show that this improved Jacobian conditioning translates to enhanced performance in practice. Our approach is simple, broadly applicable, and can be easily integrated as a drop-in replacement for a wide range of existing attention mechanisms. We validate its effectiveness across diverse transformer architectures and tasks, demonstrating consistent improvements in performance.

Deep Learning · Algorithms

Zhongnan Cai, Yingying Wang, Hui Zheng, Panwang Pan, Zixu Lin, Ge Meng, Chenxin Li, Chunming He, Jiaxin Xie, Yunlong Lin 等

Recently, deep learning-based pan-sharpening algorithms have achieved notable advancements over traditional methods. However, deep learning-based methods incur substantial computational overhead during inference, especially with large images. This excessive computational demand limits the applicability of these methods in real-world scenarios, particularly in the absence of dedicated computing devices such as GPUs and TPUs. To address these challenges, we propose Pan-LUT, a novel learnable look-up table (LUT) framework for pan-sharpening that strikes a balance between performance and computational efficiency for large remote sensing images. Our method makes it possible to process 15K$\times$15K remote sensing images on a 24GB GPU. To finely control the spectral transformation, we devise the PAN-guided look-up table (PGLUT) for channel-wise spectral mapping. To effectively capture fine-grained spatial details, we introduce the spatial details look-up table (SDLUT). Furthermore, to adaptively aggregate channel information for generating high-resolution multispectral images, we design an adaptive output look-up table (AOLUT). Our model contains fewer than 700K parameters and processes a 9K$\times$9K image in under 1 ms using one RTX 2080 Ti GPU, demonstrating significantly faster performance compared to other methods. Experiments reveal that Pan-LUT efficiently processes large remote sensing images in a lightweight manner, bridging the gap to real-world applications. Furthermore, our model surpasses SOTA methods in full-resolution scenes under real-world conditions, highlighting its effectiveness and efficiency.

Applications · Physics

Minju Jo, Woojin Cho, Uvini Balasuriya Mudiyanselage, Seungjun Lee, Noseong Park, Kookjin Lee

Scientific machine learning often involves representing complex solution fields that exhibit high-frequency features such as sharp transitions, fine-scale oscillations, and localized structures. While implicit neural representations (INRs) have shown promise for continuous function modeling, capturing such high-frequency behavior remains a challenge—especially when modeling multiple solution fields with a shared network. Prior work addressing spectral bias in INRs has primarily focused on single-instance settings, limiting scalability and generalization. In this work, we propose Global Fourier Modulation (GFM), a novel modulation technique that injects high-frequency information at each layer of the INR through Fourier-based reparameterization. This enables compact and accurate representation of multiple solution fields using low-dimensional latent vectors. Building upon GFM, we introduce PDEfuncta, a meta-learning framework designed to learn multi-modal solution fields and support generalization to new tasks. Through empirical studies on diverse scientific problems, we demonstrate that our method not only improves representational quality but also shows potential for forward and inverse inference tasks without the need for retraining.

Theory · Learning Theory

Sanjoy Dasgupta, Syamantak Kumar, Shourya Pandey, Purnamrita Sarkar

Low-precision Streaming PCA estimates the top principal component in a streaming setting under limited precision. We establish an information‐theoretic lower bound on the quantization resolution required to achieve a target accuracy for the leading eigenvector. We study Oja's algorithm for streaming PCA under linear and nonlinear stochastic quantization. The quantized variants use unbiased stochastic quantization of the weight vector and the updates. Under mild moment and spectral-gap assumptions on the data distribution, we show that a batched version achieves the lower bound up to logarithmic factors under both schemes. This leads to a nearly _dimension-free_ quantization error in the nonlinear quantization setting. Empirical evaluations on synthetic streams validate our theoretical findings and demonstrate that our low-precision methods closely track the performance of standard Oja’s algorithm.

Deep Learning · Algorithms

Devan Shah, Shlomo Fortgang, Sofiia Druchyna, Elad Hazan

We present the first provable method for identifying symmetric linear dynamical systems (LDS) with accuracy guarantees that are independent of the system’s state dimension or effective memory. Our approach builds upon recent work that represents symmetric LDSs as convolutions learnable via fixed spectral transformations. We show how to invert this representation—recovering an LDS model from its spectral transform—yielding an end-to-end convex optimization procedure. This distillation preserves predictive accuracy while enabling constant-time and constant-space inference per token, independent of sequence length. We evaluate our method, SpectraLDS, as a component in sequence prediction architectures and demonstrate that accuracy is preserved while inference efficiency is improved on tasks such as language modeling.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Phuc Tran, Van Vu, Nisheeth K. Vishnoi

A central challenge in machine learning is to understand how noise or measurement errors affect low-rank approximations, particularly in the spectral norm. This question is especially important in differentially private low-rank approximation, where one aims to preserve the top-$p$ structure of a data-derived matrix while ensuring privacy. Prior work often analyzes Frobenius norm error or changes in reconstruction quality, but these metrics can over- or under-estimate true subspace distortion. The spectral norm, by contrast, captures worst-case directional error and provides the strongest utility guarantees. We establish new high-probability spectral-norm perturbation bounds for symmetric matrices that refine the classical Eckart--Young--Mirsky theorem and explicitly capture interactions between a matrix $A \in \mathbb{R}^{n \times n}$ and an arbitrary symmetric perturbation $E$. Under mild eigengap and norm conditions, our bounds yield sharp estimates for $\| (A + E)_p - A_p \|$, where $A_p$ is the best rank-$p$ approximation of $A$, with improvements of up to a factor of $\sqrt{n}$. As an application, we derive improved utility guarantees for differentially private PCA, resolving an open problem in the literature. Our analysis relies on a novel contour bootstrapping method from complex analysis and extends it to a broad class of spectral functionals, including polynomials and matrix exponentials. Empirical results on real-world datasets confirm that our bounds closely track the actual spectral error under diverse perturbation regimes.

Probabilistic Methods · Bayesian Models and Methods

Kyurae Kim, Yian Ma, Trevor Campbell, Jacob Gardner

We prove that, given a mean-field location-scale variational family, black-box variational inference (BBVI) with the reparametrization gradient converges at a rate that is nearly independent of explicit dimension dependence. Specifically, for a $d$-dimensional strongly log-concave and log-smooth target, the number of iterations for BBVI with a sub-Gaussian family to obtain a solution $\epsilon$-close to the global optimum has a dimension dependence of $\mathrm{O}(\log d)$. This is a significant improvement over the $\mathrm{O}(d)$ dependence of full-rank location-scale families. For heavy-tailed families, we prove a weaker $\mathrm{O}(d^{2/k})$ dependence, where $k$ is the number of finite moments of the family. Additionally, if the Hessian of the target log-density is constant, the complexity is free of any explicit dimension dependence. We also prove that our bound on the gradient variance, which is key to our result, cannot be improved using only spectral bounds on the Hessian of the target log-density.

Hoang Pham, The Anh Ta, Tom Jacobs, Rebekka Burkholz, Long Tran-Thanh

Sparse neural networks promise efficiency, yet training them effectively remains a fundamental challenge. Despite advances in pruning methods that create sparse architectures, understanding why some sparse structures are better trainable than others with the same level of sparsity remains poorly understood. Aiming to develop a systematic approach to this fundamental problem, we propose a novel theoretical framework based on the theory of graph limits, particularly graphons, that characterizes sparse neural networks in the infinite-width regime. Our key insight is that connectivity patterns of sparse neural networks induced by pruning methods converge to specific graphons as networks' width tends to infinity, which encodes implicit structural biases of different pruning methods. We postulate the *Graphon Limit Hypothesis* and provide empirical evidence to support it. Leveraging this graphon representation, we derive a *Graphon Neural Tangent Kernel (Graphon NTK)* to study the training dynamics of sparse networks in the infinite width limit. Graphon NTK provides a general framework for the theoretical analysis of sparse networks. We empirically show that the spectral analysis of Graphon NTK correlates with observed training dynamics of sparse networks, explaining the varying convergence behaviours of different pruning methods. Our framework provides theoretical insights into the impact of connectivity patterns on the trainability of various sparse network architectures.

Computer Vision · Image and Video Generation

Ying Li, Chengfei Lyu, Huan Wang

Visual AutoRegressive (VAR) modeling employs a next-scale decoding paradigm that progresses from coarse structures to fine details. While enhancing fidelity and scalability, this approach challenges two fundamental assumptions of conventional dynamic inference: semantic stability (intermediate outputs approximating final results) and monotonic locality (smooth representation evolution across layers), which renders existing dynamic inference methods ineffective for VAR models. To address this challenge, we propose FreqExit, a unified training framework that enables dynamic inference in VAR without altering its architecture or compromising output quality. FreqExit is based on a key insight: high-frequency details are crucial for perceptual quality and tend to emerge only in later decoding stages. Leveraging this insight, we design targeted mechanisms that guide the model to learn more effectively through frequency-aware supervision. The proposed framework consists of three components: (1) a curriculum-based supervision strategy with progressive layer dropout and early exit loss; (2) a wavelet-domain high-frequency consistency loss that aligns spectral content across different generation steps; and (3) a lightweight self-supervised frequency-gated module that guides adaptive learning of both structural and detailed spectral components. On ImageNet 256×256, FreqExit achieves up to 2× speedup with only minor degradation, and delivers 1.3× acceleration without perceptible quality loss. This enables runtime-adaptive acceleration within a unified model, offering a favorable trade-off between efficiency and fidelity for for practical and flexible deployment.