Spectral Methods

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Overview

Spectral methods study problems by expressing data or operators in terms of eigenvalues, eigenvectors, or frequency modes. Classical approaches include Fourier and graph-Laplacian transforms, PCA and low-rank approximation, kernel methods, and polynomial filters; they are used for tasks such as clustering networks, estimating dimensions, and solving or learning partial differential equations. Newer work also uses spectral structure to design neural operators and optimizers, diagnose model behavior, and speed up large matrix computations. A central challenge is balancing efficient, robust approximations with the assumptions and information lost when choosing a fixed basis, truncating modes, or relying on noisy spectra.

Latest papers

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238 papers21 papers in the last four weeks, down 30% on the four weeks before. 0.1% of all new papers.

Apr 29, 2026

TimeMM: Time-as-Operator Spectral Filtering for Dynamic Multimodal Recommendation

Multimodal recommendation improves user modeling by integrating collaborative signals with heterogeneous item content. In real applications, user interests evolve over time and exhibit nonstationary dynamics, where different preference factors change at different rates. This challenge is amplified in multimodal settings because visual and textual cues can dominate decisions under different temporal regimes. Despite strong progress, most multimodal recommenders still rely on static interaction graphs or coarse temporal heuristics, which limits their ability to model continuous preference evolution with fine-grained temporal adaptation. To address these limitations, we propose TimeMM, a time-conditioned spectral filtering framework for dynamic multimodal recommendation. TimeMM instantiates Time-as-Operator by mapping interaction recency to a family of parametric temporal kernels that reweight edges on the user--item graph, producing component-specific representations without explicit eigendecomposition. To capture non-stationary interests, we introduce Adaptive Spectral Filtering that mixes the operator bank according to temporal context, yielding prediction-specific effective spectral responses. To account for modality-specific temporal sensitivity, we further propose Spectral-Aware Modality Routing that calibrates visual and textual contributions conditioned on the same temporal context. Finally, a ranking-space Spectral Diversity Regularization encourages complementary expert behaviors and prevents filter-bank collapse. Extensive experiments on real-world benchmarks demonstrate that TimeMM consistently outperforms state-of-the-art multimodal recommenders while maintaining linear-time scalability.

Wei Yang, Rui Zhong, Zihan Lin +4

Xiaohongshu Inc. · Beijing, China

Dynamic Graph LearningRecommender SystemsMultimodal RecommendationSpectral MethodsMultimodal Fusion
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Apr 28, 2026

Sparse Graph Learning from Sparse Data via Fiedler Number Maximization

We aim to learn a sparse and connected graph from sparse data, where the number of observations K can be substantially smaller than the signal dimension N for signals x in R^N, and the underlying distribution is unknown. In this severely ill-posed setting, we incorporate Fiedler number (the second eigenvalue of the graph Laplacian matrix that quantifies connectedness) as a robust regularization term in the sparse graph learning objective. We first develop a greedy algorithm that iteratively selects one edge globally for weakening/removal to reduce the objective, leveraging eigenvalue perturbation theorems that bound the adverse effect of an edge change to the Fiedler number. Next, we design a parallel variant, based on the Cheeger's inequality, that recursively partitions an input graph into two sub-graphs using an approximate Cheeger cut to distributedly find an optimal edge. Simulation experiments show that Fiedler number maximization robustifies sparse graph estimates, outperforming previous sparse graph learning algorithms.

Bahar Oveisgharan, Gene Cheung, Andrew Eckford

York University, Canada

Graph Structure LearningSpectral Methods
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Apr 27, 2026

Diffusion-Guided Feature Selection via Nishimori Temperature: Noise-Based Spectral Embedding

We propose Noise-Based Spectral Embedding (NBSE), a physics-informed framework for selecting informative features from high-dimensional data without greedy search. NBSE constructs a sparse similarity graph on the samples and identifies the Nishimori temperature βNβ_N the critical inverse temperature at which the Bethe Hessian becomes singular. The corresponding smallest eigenvector captures the dominant mode of an intrinsically degree-corrected diffusion process, naturally reweighting nodes to prevent hub dominance. By transposing the data matrix and applying NBSE in feature space, we obtain a one-dimensional spectral embedding that reveals groups of redundant or semantically related dimensions; balanced binning then selects one representative per group. We prove that coloured Gaussian perturbations shift βNβ_N by at most O(σˉ2)O(\barσ^2), guaranteeing robustness to measurement noise. Experiments on ImageNet embeddings from MobileNetV2 and EfficientNet-B4 show that NBSE preserves classification accuracy even under aggressive compression: on EfficientNet-B4 the accuracy drop is below 1%1\% when retaining only 30%30\% of features, outperforming ANOVA FF-test and random selection by up to 6.8%6.8\%.

Vasiliy S. Usatyuk, Denis A. Sapozhnikov, Sergey I. Egorov

R&D Department T8 LLC Moscow, Russia · Department of Computer Science South-West State University Kursk, Russia

Feature SelectionDimensionality ReductionSpectral MethodsSpectral Graph Embedding
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Apr 25, 2026

Learning Curves and Benign Overfitting of Spectral Algorithms in Large Dimensions

Existing large-dimensional theory for spectral algorithms resolves either the optimally tuned point or the interpolation limit, but leaves the under-regularized regime unexplored. We study the learning curve and benign overfitting of spectral algorithms in the large-dimensional setting where the sample size and dimension are of comparable order, i.e., n≍dγn \asymp d^γ for some γ>0γ>0. We first consider inner-product kernels on the sphere Sd−1\mathbb{S}^{d-1} and establish a sharp asymptotic characterization of the excess risk across the full regularization path under various source conditions s≥0s \geq 0, where ss measures the relative smoothness of the regression function. Our results reveal that the learning curve is not simply U-shaped but instead consists of three distinct regimes: over-regularized, under-regularized, and interpolation regimes. This characterization allows us to fully capture the benign overfitting phenomenon, demonstrating that benign overfitting arises consistently across both the under-regularized and interpolation regimes whenever ss is positive but no larger than a critical threshold. We further show that, in the sufficiently regularized regime, the kernel learning curve is recovered by an associated sequence model. Finally, we extend the learning-curve analysis to large-dimensional KRR for a class of kernels on general domains in Rd\mathbb{R}^d whose low-degree eigenspaces satisfy spectral-scaling and hyper-contractivity conditions.

Weihao Lu, Qian Lin, Yingcun Xia +1

Department of Statistics and Data Science, National University of Singapore. · Department of Statistics and Data Science, Tsinghua University.

Kernel Ridge RegressionBenign OverfittingLearning CurvesSpectral Methods
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Apr 24, 2026

A Brain-Inspired Deep Separation Network for Single Channel Raman Spectra Unmixing

Raman spectra obtained in real world applications are often a noisy combination of several spectra of various substances in a tested sample. Unmixing such spectra into individual components corresponding to each of the substances is of great value and has been a longstanding challenge in Raman spectroscopy. Existing unmixing methods are predominantly designed to invert an overdetermined mixed model and therefore require multiple mixed spectra as input. However, open domain and/or non-cooperative detection applications in Raman spectroscopy such as controlled substance detection, call for single-channel solutions which can identify individual components from thousands of candidates by analyzing only a single noisy mixed spectrum. To our knowledge, sparse regression is the only existing solution which can cope with this scenario, yet it has very low tolerance to noises and can hardly be applicable in practice. To address these limitations, we introduce a novel neural approach for single-channel Raman spectrum unmixing inspired by speech separation. It aims at solving underdetermined systems and can decompose a noisy mixed spectrum from a library of thousands of components (substances). The core of our method is a deep separation neural network (RSSNet) which takes a mixed spectrum as input and outputs spectra of pure components. We created two synthetic datasets of single-channel Raman spectra unmixing and demonstrated feasibility and superiority of RSSNet on these datasets (outperform competing methods by >4dB). Furthermore, we verified that RSSNet, trained solely on synthetic data, can successfully unmix real-world mixed spectra of mixtures of mineral powders, exhibiting strong generalization. Our approach represents a new paradigm for Raman unmixing and enables new possibilities for fast detection of Raman mixtures.

Gaoruishu Long, Jinchao Liu, Bo Liu +2

College of Artificial Intelligence, Nankai University, Tianjin, China · Guangdong Laboratory of Chemistry and Fine Chemical Industry, Guangdong, China · Smekal Tech (Shantou) Ltd. +1

Raman SpectroscopySpectral MethodsSpectral Unmixing
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Apr 22, 2026

Spectral Embeddings Leak Graph Topology: Theory, Benchmark, and Adaptive Reconstruction

Graph Neural Networks (GNNs) excel on relational data, but standard benchmarks unrealistically assume the graph is centrally available. In practice, settings such as Federated Graph Learning, distributed systems, and privacy-sensitive applications involve graph data that are localized, fragmented, noisy, and privacy-leaking. We present a unified framework for this setting. We introduce LoGraB (Local Graph Benchmark), which decomposes standard datasets into fragmented benchmarks using three strategies and four controls: neighborhood radius dd, spectral quality kk, noise level σσ, and coverage ratio pp. LoGraB supports graph reconstruction, localized node classification, and inter-fragment link prediction, with Island Cohesion. We propose AFR (Adaptive Fidelity-driven Reconstruction), a method for noisy spectral fragments. AFR scores patch quality via a fidelity measure combining a gap-to-truncation stability ratio and structural entropy, then assembles fragments using RANSAC-Procrustes alignment, adaptive stitching, and Bundle Adjustment. Rather than forcing a single global graph, AFR recovers large faithful islands. We prove heat-kernel edge recovery under a separation condition, Davis--Kahan perturbation stability, and bounded alignment error. We establish a Spectral Leakage Proposition: under a spectral-gap assumption, polynomial-time Bayesian recovery is feasible once enough eigenvectors are shared, complementing AFR's deterministic guarantees. Experiments on nine benchmarks show that LoGraB reveals model strengths and weaknesses under fragmentation, AFR achieves the best F1 on 7/9 datasets, and under per-embedding (ε,δ)(ε,δ)-Gaussian differential privacy, AFR retains 75% of its undefended F1 at ε=2ε=2. Our anonymous code is available at https://anonymous.4open.science/r/JMLR_submission

Thinh Nguyen-Cong, Truong-Son Hy, Thang N. Dinh

Department of Computer Science Virginia Commonwealth University Richmond, VA, USA · Department of Computer Science The University of Alabama at Birmingham Birmingham, AL, USA

Graph ReconstructionSpectral MethodsSpectral Graph Embedding
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Apr 22, 2026

Improved large-scale graph learning through ridge spectral sparsification

Graph-based techniques and spectral graph theory have enriched the field of machine learning with a variety of critical advances. A central object in the analysis is the graph Laplacian L, which encodes the structure of the graph. We consider the problem of learning over this Laplacian in a distributed streaming setting, where new edges of the graph are observed in real time by a network of workers. In this setting, it is hard to learn quickly or approximately while keeping a distributed representation of L. To address this challenge, we present a novel algorithm, GSQUEAK, which efficiently sparsifies the Laplacian by maintaining a small subset of effective resistances. We show that our algorithm produces sparsifiers with strong spectral approximation guarantees, all while processing edges in a single pass and in a distributed fashion.

Daniele Calandriello, Ioannis Koutis, Alessandro Lazaric +1

SequeL team, INRIA Lille - Nord Europe, France · LCSL, IIT, Italy · New Jersey Institute of Technology, USA +1

Graph SparsificationSpectral MethodsStreaming Algorithms
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Apr 22, 2026

Analysis of Nystrom method with sequential ridge leverage scores

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix K_t. To avoid storing the entire matrix K_t, Nystrom methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate KRR solution on the reconstructed matrix. The chosen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, recent works show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarantees for the approximation. While exact RLSs are as difficult to compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-ESTIMATE algorithm, that incrementally computes the RLSs estimates. INK-ESTIMATE maintains a small sketch of K_t, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximation guarantees on the distance between the true kernel matrix and its approximation, and on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.

Daniele Calandriello, Alessandro Lazaric, Michal Valko

SequeL team INRIA Lille - Nord Europe

Kernel Ridge RegressionKernel RegressionNyström MethodRandomized SketchingSpectral MethodsRidge Regression+1
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Apr 21, 2026

Achieving the Kesten-Stigum bound in the non-uniform hypergraph stochastic block model

We study the community detection problem in the non-uniform hypergraph stochastic block model (HSBM), where hyperedges of varying sizes coexist. This setting captures higher-order and multi-view interactions and raises a fundamental question: can multiple uniform hypergraph layers below the detection threshold be combined to enable weak recovery? We answer this question by establishing a Kesten--Stigum-type bound for weak recovery in a general class of non-uniform HSBMs with rr blocks, generated according to multiple symmetric probability tensors. In the case r=2r=2, we show that weak recovery is possible whenever the sum of the signal-to-noise ratios across all uniform hypergraph layers exceeds one, thereby confirming the positive part of a conjecture in (Chodrow et al., 2023). Moreover, we provide a polynomial-time spectral algorithm that achieves this threshold via an optimally weighted non-backtracking operator. For the unweighted non-backtracking matrix, our spectral method attains a different algorithmic threshold, also conjectured in (Chodrow et al., 2023). Our approach develops a spectral theory for weighted non-backtracking operators on non-uniform hypergraphs, including a precise characterization of outlier eigenvalues and eigenvector overlaps. We introduce a novel Ihara--Bass formula tailored to weighted non-uniform hypergraphs, which yields an efficient low-dimensional representation and leads to a provable spectral reconstruction algorithm. Taken together, these results provide a principled and computationally efficient approach to clustering in non-uniform hypergraphs, and highlight the role of optimal weighting in aggregating heterogeneous higher-order interactions.

Manuel Fernandez, Ludovic Stephan, Yizhe Zhu

Department of Mathematics, University of Southern California · Univ Rennes, Ensai, CNRS, CREST-UMR 9194, F-35000 Rennes, France

Spectral ClusteringStochastic Block ModelSpectral Methods
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Apr 20, 2026

Predicting LLM Compression Degradation from Spectral Statistics

Matrix-level low-rank compression is a promising way to reduce the cost of large language models, but running compression and evaluating the resulting models on language tasks can be prohibitively expensive. Can compression-induced degradation be predicted before committing to this compute? We systematically analyze the Qwen3 and Gemma3 model families across four representative low-rank compression methods: vanilla SVD, two ASVD variants, and SVD-LLM. We find that stable rank and information density, measured in bits per parameter, dominate performance degradation. The interaction term γ⋅ρˉsγ\cdot \barρ_s, defined as compression ratio times stable rank, is a robust predictor of accuracy degradation, achieving leave-one-out cross-validation Pearson correlations of 0.8900.890 for attention layers and 0.8390.839 for MLP layers. We provide theoretical intuition for why this predictor succeeds by connecting it to standard SVD truncation bounds and error composition mechanisms in transformer layers. These findings enable a predict-then-compress workflow: compute γ⋅ρˉsγ\cdot \barρ_s from weights, estimate degradation, and invest compute only in desirable configurations.

Mingxue Xu

Department of Electrical and Electronic Engineering, Imperial College London, London, United Kingdom

LLM CompressionLow-Rank Matrix DecompositionLow-Rank CompressionLow-Rank ApproximationSpectral Methods
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Apr 19, 2026

Spectral Forensics of Diffusion Attention Graphs for Copy-Move Forgery Detection

Copy-move forgery, where a region within an image is duplicated to hide or fabricate content, remains a persistent threat to visual media integrity. We introduce GraphSpecForge, a training-free framework that detects copy-move forgery by analysing the spectral structure of attention graphs from a pretrained Stable Diffusion U-Net. Our central insight is that copy-move manipulation induces approximate subgraph duplication in the self-attention graph, leading to measurable spectral redistribution in the normalized graph Laplacian. We formalise this link with perturbation-based arguments and build an image-level anomaly detector using Wasserstein distances between per-image Laplacian spectra and an authentic reference distribution. We evaluate GraphSpecForge on four copy-move benchmarks without forgery-specific retraining. On RecodAI-LUC (5,128 images), our best configuration achieves AUROC = 0.606 (95% CI: 0.580-0.638; permutation p = 0.005), and the normalized Laplacian outperforms raw attention spectra by +0.057 AUROC. On MICC-F220, CoMoFoD, and COVERAGE, the same pipeline attains AUROCs of 0.752, 0.774, and 0.673, respectively; on CoMoFoD it also reaches AUPRC = 0.833, balanced accuracy = 0.712, MCC = 0.499, and TPR@1%FPR = 32.5%. Additional ablation and falsification experiments confirm the signal's specificity and sensitivity to manipulation strength, while null-graph controls rule out trivial-statistic explanations.

H. M. Shadman Tabib, Tasriad Ahmed Tias, Nafis Tahmid

Department of Computer Science and Engineering Bangladesh University of Engineering and Technology (BUET) Dhaka, Bangladesh

Image Forgery DetectionDigital Image ForensicsSpectral Methods
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Apr 18, 2026

Spectral Analysis of Fake News Propagation

The propagation structure of fake news has been shown to be an important cue for detecting it; yet, existing propagation-based fake news detection methods have mainly relied on ad hoc topological features, and a unified view of cascade patterns is still lacking. To address this, we study news propagation from a spectral view by connecting graph spectra to propagation-related structural properties through rigorous spectral bounds. In particular, we introduce several new bounds and integrate them with existing ones into a unified spectral representation of information propagation. We then use these spectral bounds for downstream classification and design a discrete structural optimization framework to interpret learned propagation patterns. For efficient optimization, we rely on a first-order perturbation approximation and consider both score-guided and bound-guided objectives. Experiments on real-world data reveal meaningful spectral differences between fake and real news, competitive classification performance from spectral bounds, and interpretable evolution trajectories from structural optimization. The findings demonstrate the value of spectral analysis for understanding and modeling news propagation.

Weibin Cai, Reza Zafarani

Data Lab, Department of EECS, Syracuse University, New York, USA.

Social Media AnalysisFake News DetectionSpectral Methods
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Apr 18, 2026

Convergence theory for Hermite approximations under adaptive coordinate transformations

Recent work has shown that parameterizing and optimizing coordinate transformations using normalizing flows, i.e., invertible neural networks, can significantly accelerate the convergence of spectral approximations. We present the first error estimates for approximating functions using Hermite expansions composed with adaptive coordinate transformations. Our analysis establishes an equivalence principle: approximating a function ff in the span of the transformed basis is equivalent to approximating the pullback of ff in the span of Hermite functions. This allows us to leverage the classical approximation theory of Hermite expansions to derive error estimates in transformed coordinates in terms of the regularity of the pullback. We present an example demonstrating how a nonlinear coordinate transformation can enhance the convergence of Hermite expansions. Focusing on smooth functions decaying along the real axis, we construct a monotone transport map that aligns the decay of the target function with the Hermite basis. This guarantees spectral convergence rates for the corresponding Hermite expansion. Our analysis provides theoretical insight into the convergence behavior of adaptive Hermite approximations based on normalizing flows, as recently explored in the computational quantum physics literature.

Yahya Saleh

Spectral Methods
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Apr 17, 2026

Spectral Kernel Dynamics for Planetary Surface Graphs: Distinction Dynamics and Topological Conservation

The spectral kernel field equation R[k] = T[k] lacks a conservation-law analog. We prove (i) the fixed-point flow is strictly volume-expanding (tr DF > 0), precluding automatic conservation, and (ii) the conservation deficit per mode equals the Hessian stability margin exactly: D_m = -Delta'. Closing the deficit requires a scene-side compensating contribution, which we formalise as the distinction dynamics equation dc/dt = G[c, h_t], with MaxCal-optimal realisation G_opt. On fixed-topology 3D surface graphs we derive a conditional topology-preserving compression theorem: retaining k >= beta_0 + beta_1 modes (under a spectral-ordering assumption) preserves all Betti-number charges; we include a worked short-cycle counterexample (figure-eight) calibrating when the assumption fails. A triple necessary spectral diagnostic -- Fiedler-mode concentration, elevated curl energy, anomalous beta_1 -- is derived for planetary drainage networks at O(N) cost. Two internal real-data sequences serve as preliminary consistency checks; full benchmarks and adaptive-topology extensions are deferred.

Jnaneshwar Das

School of Earth and Space Exploration, Arizona State University

Dynamical SystemsSpectral MethodsTopological Data Analysis
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Apr 15, 2026

Spectral Thompson sampling

Thompson Sampling (TS) has attracted a lot of interest due to its good empirical performance, in particular in the computational advertising. Though successful, the tools for its performance analysis appeared only recently. In this paper, we describe and analyze SpectralTS algorithm for a bandit problem, where the payoffs of the choices are smooth given an underlying graph. In this setting, each choice is a node of a graph and the expected payoffs of the neighboring nodes are assumed to be similar. Although the setting has application both in recommender systems and advertising, the traditional algorithms would scale poorly with the number of choices. For that purpose we consider an effective dimension d, which is small in real-world graphs. We deliver the analysis showing that the regret of SpectralTS scales as dsqrt(T ln N) with high probability, where T is the time horizon and N is the number of choices. Since a dsqrt(T ln N) regret is comparable to the known results, SpectralTS offers a computationally more efficient alternative. We also show that our algorithm is competitive on both synthetic and real-world data.

Tomas Kocak, Michal Valko, Remi Munos +1

SequeL team, INRIA Lille - Nord Europe, France · SequeL team, INRIA Lille, France · Microsoft Research NE, USA +1

Thompson SamplingGraph BanditsSpectral Methods
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Apr 13, 2026

STS-Mixer: Spatio-Temporal-Spectral Mixer for 4D Point Cloud Video Understanding

4D point cloud videos capture rich spatial and temporal dynamics of scenes which possess unique values in various 4D understanding tasks. However, most existing methods work in the spatiotemporal domain where the underlying geometric characteristics of 4D point cloud videos are hard to capture, leading to degraded representation learning and understanding of 4D point cloud videos. We address the above challenge from a complementary spectral perspective. By transforming 4D point cloud videos into graph spectral signals, we can decompose them into multiple frequency bands each of which captures distinct geometric structures of point cloud videos. Our spectral analysis reveals that the decomposed low-frequency signals capture more coarse shapes while high-frequency signals encode more fine-grained geometry details. Building on these observations, we design Spatio-Temporal-Spectral Mixer (STS-Mixer), a unified framework that mixes spatial, temporal, and spectral representations of point cloud videos. STS-Mixer integrates multi-band delineated spectral signals with spatiotemporal information to capture rich geometries and temporal dynamics, while enabling fine-grained and holistic understanding of 4D point cloud videos. Extensive experiments show that STS-Mixer achieves superior performance consistently across multiple widely adopted benchmarks on both 3D action recognition and 4D semantic segmentation tasks. Code and models are available at https://github.com/Vegetebird/STS-Mixer.

Wenhao Li, Xueying Jiang, Gongjie Zhang +3

Nanyang Technological University · Alibaba Group · Zhejiang University of Technology +1

Point Cloud Learning4D Scene UnderstandingSpectral MethodsHuman Activity Recognition
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Apr 10, 2026

Spectral Kernel Dynamics via Maximum Caliber: Fixed Points, Geodesics, and Phase Transitions

We derive a closed-form geometric functional for kernel dynamics on finite graphs by applying the Maximum Caliber (MaxCal) variational principle to the spectral transfer function h(lambda) of the graph Laplacian eigenbasis. The main result is that the MaxCal stationarity condition decouples into N one-dimensional problems with explicit solution: h*(lambda_l) = h_0(lambda_l) exp(-1 - T_l[h*]), yielding self-consistent (fixed-point) kernels via exponential tilting (Corollary 1), log-linear Fisher-Rao geodesics (Corollary 2), a diagonal Hessian stability criterion (Corollary 3), and an l^2_+ isometry for the spectral kernel space (Proposition 3). The spectral entropy H[h_t] provides a computable O(N) early-warning signal for network-structural phase transitions (Remark 7). All claims are numerically verified on the path graph P_8 with a Gaussian mutual-information source, using the open-source kernelcal library. The framework is grounded in a structural analogy with Einstein's field equations, used as a guiding template rather than an established equivalence; explicit limits are stated in Section 6.

Jnaneshwar Das

School of Earth and Space Exploration, Arizona State University

Kernel MethodsSpectral MethodsInformation GeometryPhase Transitions
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Apr 8, 2026

The Lifecycle of the Spectral Edge: From Gradient Learning to Weight-Decay Compression

We decompose the spectral edge -- the dominant direction of the Gram matrix of parameter updates -- into its gradient and weight-decay components during grokking in two sequence tasks (Dyck-1 and SCAN). We find a sharp two-phase lifecycle: before grokking the edge is gradient-driven and functionally active; at grokking, gradient and weight decay align, and the edge becomes a compression axis that is perturbation-flat yet ablation-critical (>4000x more impactful than random directions). Three universality classes emerge (functional, mixed, compression), predicted by the gap flow equation. Nonlinear probes show information is re-encoded, not lost (MLP R2=0.99R^2=0.99 where linear R2=0.86R^2=0.86), and removing weight decay post-grok reverses compression while preserving the algorithm.

Yongzhong Xu

Neural Network Training DynamicsGrokkingWeight DecayNeural Network CompressionSpectral Methods
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Apr 5, 2026

Subspace Control: Turning Constrained Model Steering into Controllable Spectral Optimization

Foundation models, such as large language models (LLMs), are powerful but often require customization before deployment to satisfy practical constraints such as safety, privacy, and task-specific requirements, leading to "constrained" optimization problems for model steering and adaptation. However, solving such problems remains largely underexplored and is particularly challenging due to interference between the primary objective and constraint objectives during optimization. In this paper, we propose a subspace control framework for constrained model training. Specifically, (i) we first analyze, from a model merging perspective, how spectral cross-task interference arises and show that it can be resolved via a one-shot solution that orthogonalizes the merged subspace; (ii) we establish a connection between this solution and gradient orthogonalization in the spectral optimizer Muon; and (iii) building on these insights, we introduce SIFT (spectral interference-free training), which leverages a localization scheme to selectively intervene during optimization, enabling controllable updates that mitigate objective-constraint conflicts. We evaluate SIFT across four representative applications: (a) machine unlearning, (b) safety alignment, (c) text-to-speech adaptation, and (d) hallucination mitigation. Compared to both control-based and control-free baselines, SIFT consistently achieves substantial and robust performance improvements across all tasks. Code is available at https://github.com/OPTML-Group/SIFT.

Yancheng Huang, Changsheng Wang, Chongyu Fan +7

OPTML, Michigan State University · MIT-IBM Watson AI Lab, IBM Research · University of Minnesota +2

Spectral OptimizationSpectral Methods
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Apr 5, 2026

Spectral Path Regression: Directional Chebyshev Harmonics for Interpretable Tabular Learning

Classical approximation bases such as Chebyshev polynomials provide principled and interpretable representations, but their multivariate tensor-product constructions scale exponentially with dimension and impose axis-aligned structure that is poorly matched to real tabular data. We address this by replacing tensorised oscillations with directional harmonic modes of the form cos⁡(m⊤arccos⁡(x))\cos(\mathbf{m}^{\top}\arccos(\mathbf{x})), which organise multivariate structure by direction in angular space rather than by coordinate index. This representation yields a discrete spectral regression model in which complexity is controlled by selecting a small number of structured frequency vectors (spectral paths), and training reduces to a single closed-form ridge solve with no iterative optimisation. Experiments on standard continuous-feature tabular regression benchmarks show that the resulting models achieve accuracy competitive with strong nonlinear baselines while remaining compact, computationally efficient, and explicitly interpretable through analytic expressions of learned feature interactions.

Milo Coombs

Tabular RegressionSpectral MethodsFrequency-Domain Feature Learning
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Apr 4, 2026

Polynomial-Time Optimal Group Selection via the Double-Commutator Eigenvalue Problem

The algebraic diversity framework generalizes temporal averaging over multiple observations to algebraic group action on a single observation for second-order statistical estimation. The central open problem in this framework is group selection\textit{group selection}: given an MM-dimensional observation with unknown covariance structure, find the finite group whose spectral decomposition best matches the covariance. Naive enumeration of all subgroups of the symmetric group SMS_M requires exponential time in MM. We prove that this combinatorial problem reduces to a generalized eigenvalue problem derived from the double commutator of the covariance matrix, yielding a polynomial-time algorithm with complexity O(d2M2+d3)O(d^2M^2 + d^3), where dd is the dimension of a generator basis. The minimum eigenvector of the double-commutator matrix directly constructs the optimal group generator in closed form, with no iterative optimization. The reduction is exact: the double-commutator minimum eigenvalue is zero if and only if the optimal generator lies in the span of the basis, and its magnitude provides a certifiable optimality gap when it does not. This problem does not appear in the standard catalogs of computational complexity (Garey and Johnson, 1979) and represents a new class linking group theory, matrix analysis, and statistical estimation. We establish connections to independent component analysis (JADE), structured matrix nearness problems, and simultaneous matrix diagonalization, and we show that the double-commutator formulation is the unique approach that is simultaneously polynomial-time, closed-form, and certifiable. We extend the framework to non-Abelian symmetry recovery via a Sequential GEVP with deflation, and add two identifiability theorems characterizing the commutant-lattice ambiguity and the dichotomy on whether Aut(R)\mathrm{Aut}(\mathbf{R}) recovers a generative subgroup or only a supergroup.

Mitchell A. Thornton

Richardson, TX 75080 USA

Covariance EstimationSpectral Methods
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Apr 4, 2026

Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations

We establish that temporal averaging over multiple observations is the degenerate case of algebraic group action with the trivial group G={e}G=\{e\}. A General Replacement Theorem proves that a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation. The Trivial Group Embedding Theorem proves that the sample covariance is the accumulation of trivial-group estimates, with variance governed by a (G,L)(G,L) continuum as 1/(∣G∣⋅L)1/(|G|\cdot L). The processing gain 10log⁡10(M)10\log_{10}(M) dB equals the classical beamforming gain, establishing that this gain is a property of group order, not sensor count. The DFT, DCT, and KLT are unified as group-matched special cases. We conjecture a General Algebraic Averaging Theorem extending these results to arbitrary statistics, with variance governed by the effective group order deffd_{\mathrm{eff}}. Monte Carlo experiments on the first four sample moments across five group types confirm the conjecture to four-digit precision. The framework exploits the structurestructure of information (representation-theoretic symmetry of the data object) rather than the content, complementing Shannon's theory. Five applications are demonstrated: single-snapshot MUSIC, massive MIMO, single-pulse waveform classification, graph signal processing, and analysis of transformer LLMs. Techniques for blind group matching are described.

Mitchell A. Thornton

Richardson, TX 75080 USA

Spectral Methods
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Apr 3, 2026

The Spectral Lifecycle of Transformer Training: Transient Compression Waves, Persistent Spectral Gradients, and the Q/K--V Asymmetry

We present the first systematic study of weight matrix singular value spectra during transformer pretraining, tracking full SVD decompositions of every weight matrix at 25-step intervals across three model scales (30M--285M parameters). We discover three phenomena: (1)~Transient Compression Waves: stable rank compression propagates as a traveling wave from early to late layers, creating a dramatic gradient that peaks early then reverses -- late layers eventually over-compress past early layers. (2)~Persistent Spectral Gradients: the power-law exponent~αα develops a permanent depth gradient forming a non-monotonic inverted-U in deeper models, with peaks shifting toward earlier layers as depth increases. (3)~Q/K--V Functional Asymmetry: value/output projections compress uniformly while query/key projections carry the full depth-dependent dynamics. The dissociation between transient compression and persistent spectral shape reveals that rank and spectral shape encode fundamentally different information about training. We formalize this as a two-timescale dynamical model and derive scaling laws (Δα∝L0.26Δα\propto L^{0.26}, R2=0.99R^2{=}0.99). We validate on nine models across three families (custom, GPT-2, Pythia; 30M--1B parameters; 8--36 layers), demonstrate that αα predicts layer importance (ρ=0.69ρ{=}0.69--0.840.84, p<0.02p{<}0.02), and show that spectral-guided pruning outperforms Last-N heuristics by 1.1×1.1{\times}--3.6×3.6{\times} across seven models in two families (GPT-2 124M--774M, Pythia 160M--1B), with worst-vs-best gaps up to 23.7×23.7{\times} confirming the causal role of spectral structure.

Yi Liu

Transformer Training DynamicsTransformerNeural Network Training DynamicsSpectral Methods
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Apr 2, 2026

A Spectral Decomposition Framework for Multiscale Nonlinear Dimensionality Reduction

Dimensionality reduction (DR) involves two longstanding trade-offs. First, preserving local neighborhoods can come at the cost of global structure. Neighbor embedding methods such as t-SNE and UMAP prioritize local similarity preservation but do not explicitly constrain global organization, whereas standard spectral methods such as Laplacian Eigenmaps capture smooth, coarse-scale graph structure but offer limited flexibility to depict finer local structure. Second, the flexibility of nonlinear DR methods often comes at the cost of analytical transparency. Many methods do not explicitly reveal how high-dimensional structure produces patterns in the embedding. We introduce SDMP (Spectral Decomposition for Multiscale Projection), a nonlinear DR framework built on an explicit spectral decomposition. In this formulation, each embedding dimension is expressed as a weighted combination of Laplacian eigenvectors derived from a neighborhood graph, with the weights learned via a UMAP-style cross-entropy objective. By progressively expanding the spectral subspace to capture increasingly fine graph structure, SDMP produces a sequence of embeddings, making the evolving balance between global organization and local detail explicit, controllable, and inspectable. The explicit decomposition also reveals which spectral scales shape the overall embedding and how individual eigenvectors influence point positions. Quantitative evaluations on synthetic, image, and single-cell data show competitive local and global structure preservation, while case studies illustrate how the decomposition supports interpretation of clusters and developmental trajectories across spectral scales.

Zeyang Huang, Angelos Chatzimparmpas, Thomas Höllt +1

Linköping University, Sweden · Utrecht University, The Netherlands · University of Bergen, Norway +1

Dimensionality ReductionSpectral Methods
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Apr 1, 2026

Generalization Bounds for Spectral GNNs via Fourier Domain Analysis

Spectral graph neural networks learn graph filters, but their behavior with increasing depth and polynomial order is not well understood. We analyze these models in the graph Fourier domain, where each layer becomes an element-wise frequency update, separating the fixed spectrum from trainable parameters and making depth and order explicit. In this setting, we show that Gaussian complexity is invariant under the Graph Fourier Transform, which allows us to derive data-dependent, depth, and order-aware generalization bounds together with stability estimates. In the linear case, our bounds are tighter, and on real graphs, the data-dependent term correlates with the generalization gap across polynomial bases, highlighting practical choices that avoid frequency amplification across layers.

Vahan A. Martirosyan, Daniele Malitesta, Hugues Talbot +2

Université Paris-Saclay, CentraleSupélec, Inria, France · LTCI, Télécom Paris, Institut Polytechnique de Paris, France

Generalization BoundsNeural Network GeneralizationSpectral Graph ConvolutionSpectral Methods
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Mar 30, 2026

Foundations of Polar Linear Algebra

This work revisits operator learning from a spectral perspective by introducing Polar Linear Algebra, a structured framework based on polar geometry that combines a linear radial component with a periodic angular component. Starting from this formulation, we define the associated operators and analyze their spectral properties. As a proof of feasibility, the framework is evaluated on a canonical benchmark (MNIST). Despite the simplicity of the task, the results demonstrate that polar and fully spectral operators can be trained reliably, and that imposing self-adjoint-inspired spectral constraints improves stability and convergence. Beyond accuracy, the proposed formulation leads to a reduction in parameter count and computational complexity, while providing a more interpretable representation in terms of decoupled spectral modes. By moving from a spatial to a spectral domain, the problem decomposes into orthogonal eigenmodes that can be treated as independent computational pipelines. This structure naturally exposes an additional dimension of model parallelization, complementing existing parallel strategies without relying on ad-hoc partitioning. Overall, the work offers a different conceptual lens for operator learning, particularly suited to problems where spectral structure and parallel execution are central.

Giovanni Guasti

Operator LearningSpectral Methods
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Mar 30, 2026

NeiGAD: Augmenting Graph Anomaly Detection via Spectral Neighbor Information

Graph anomaly detection (GAD) aims to identify irregular nodes or structures in attributed graphs. Neighbor information, which reflects both structural connectivity and attribute consistency with surrounding nodes, is essential for distinguishing anomalies from normal patterns. Although recent graph neural network (GNN)-based methods incorporate such information through message passing, they often fail to explicitly model its effect or interaction with attributes, limiting detection performance. This work introduces NeiGAD, a novel plug-and-play module that captures neighbor information through spectral graph analysis. Theoretical insights demonstrate that eigenvectors of the adjacency matrix encode local neighbor interactions and progressively amplify anomaly signals. Based on this, NeiGAD selects a compact set of eigenvectors to construct efficient and discriminative representations. Experiments on eight real-world datasets show that NeiGAD consistently improves detection accuracy and outperforms state-of-the-art GAD methods. These results demonstrate the importance of explicit neighbor modeling and the effectiveness of spectral analysis in anomaly detection. Code is available at: https://github.com/huafeihuang/NeiGAD.

Qing Qing, Huafei Huang, Mingliang Hou +2

College of Computer Science and Technology, Jilin University, Changchun 130012, China · School of Computer Science and Information Technology, Adelaide University, SA 5095, Australia · Guangdong Institute of Smart Education, Jinan University, Guangzhou 510632, China +1

Graph-Based Anomaly DetectionAnomaly DetectionGraph Neural NetworksSpectral Methods
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Mar 29, 2026

Spectral Signatures of Data Quality: Eigenvalue Tail Index as a Diagnostic for Label Noise in Neural Networks

We investigate whether spectral properties of neural network weight matrices can predict test accuracy. Under controlled label noise variation, the tail index alpha of the eigenvalue distribution at the network's bottleneck layer predicts test accuracy with leave-one-out R^2 = 0.984 (21 noise levels, 3 seeds per level), far exceeding all baselines: the best conventional metric (Frobenius norm of the optimal layer) achieves LOO R^2 = 0.149. This relationship holds across three architectures (MLP, CNN, ResNet-18) and two datasets (MNIST, CIFAR-10). However, under hyperparameter variation at fixed data quality (180 configurations varying width, depth, learning rate, and weight decay), all spectral and conventional measures are weak predictors (R^2 < 0.25), with simple baselines (global L_2 norm, LOO R^2 = 0.219) slightly outperforming spectral measures (tail alpha, LOO R^2 = 0.167). We therefore frame the tail index as a data quality diagnostic: a powerful detector of label corruption and training set degradation, rather than a universal generalization predictor. A noise detector calibrated on synthetic noise successfully identifies real human annotation errors in CIFAR-10N (9% noise detected with 3% error). We identify the information-processing bottleneck layer as the locus of this signature and connect the observations to the BBP phase transition in spiked random matrix models. We also report a negative result: the level spacing ratio <r> is uninformative for weight matrices due to Wishart universality.

Matthew Loftus

Independent researcher

Spectral MethodsNoisy-Label Learning
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Mar 27, 2026

Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory

Spectral optimizers such as Muon have recently shown strong empirical performance in large-scale language model training, but the source and extent of their advantage remain poorly understood. We study this question through the linear associative memory problem, a tractable model for factual recall in transformer-based models. In particular, we go beyond orthogonal embeddings and consider Gaussian inputs and outputs, which allows the number of stored associations to greatly exceed the embedding dimension. Our main result sharply characterizes the recovery rates of one step of Muon, SGD, and Newton's method on the logistic regression loss under a power law frequency distribution. We show that the storage capacity of Muon significantly exceeds that of SGD, and even matches Newton's method while only using first-order information. Moreover, Muon saturates at a larger critical batch size. We further analyze the multi-step dynamics under a thresholded gradient approximation and show that Muon achieves a substantially faster initial recovery rate than SGD, while both methods eventually converge to the information-theoretic limit at comparable speeds. Experiments on synthetic tasks validate the predicted scaling laws. Our analysis provides a quantitative understanding of the signal amplification of spectral preconditioners and lays the groundwork for establishing scaling laws across more practical language modeling tasks and optimizers.

Juno Kim, Eshaan Nichani, Denny Wu +2

UC Berkeley · Princeton University · New York University +1

Spectral OptimizationNeural Network OptimizationMuon OptimizerSpectral MethodsAssociative Memory
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Mar 25, 2026

Spectral methods: crucial for machine learning, natural for quantum computers?

This article presents an argument for why quantum computers could unlock new methods for machine learning. We argue that spectral methods, in particular those that learn, regularise, or otherwise manipulate the Fourier spectrum of a machine learning model, are often natural for quantum computers. For example, if a generative machine learning model is represented by a quantum state, the Quantum Fourier Transform allows us to manipulate the Fourier spectrum of the state using the entire toolbox of quantum routines, an operation that is usually prohibitive for classical models. At the same time, spectral methods are surprisingly fundamental to machine learning: A spectral bias has recently been hypothesised to be the core principle behind the success of deep learning; support vector machines have been known for decades to regularise in Fourier space, and convolutional neural nets build filters in the Fourier space of images. Could, then, quantum computing open fundamentally different, much more direct and resource-efficient ways to design the spectral properties of a model? We discuss this potential in detail here, hoping to stimulate a direction in quantum machine learning research that puts the question of ``why quantum?'' first.

Vasilis Belis, Joseph Bowles, Rishabh Gupta +2

Xanadu Quantum Technologies Inc., Toronto, Ontario, M5G 2C8, Canada

Quantum Machine LearningSpectral Methods
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