Spectral Methods

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17 papers in the last four weeks, up 42% on the four weeks before. 0.2% of all new papers.

Jul 13Week of Sep 28

Latest papers 170

Apr 17, 2026math.DS

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 Systems Spectral Methods Topological Data Analysis

Apr 4, 2026cs.LG

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\}

. 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)

continuum as

1/(|G|\cdot L)

. The processing gain

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

d_{\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

structure

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

Apr 2, 2026cs.LG

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 Reduction Spectral Methods

Mar 20, 2026stat.ML

Model Selection and Parameter Estimation for Multidimensional Gaussian Mixture Models with a Common Covariance Matrix

We study model-order selection and component-mean estimation for multidimensional Gaussian mixture models with a known common covariance matrix. Using empirical characteristic-function measurements, we construct Fourier covariance matrices whose population counterparts have rank equal to the number of mixture components. We establish a minimax lower bound showing that distinguishing a separated

k

-component mixture from the class of

(k-1)

-component mixtures requires

Ω(Δ^{-(4k-4)})

samples. We then develop an oracle spectral-thresholding estimator with a sufficient sample size of order

Δ^{-(8k-8)}

for fixed

k

, together with a practical singular-value-ratio estimator. Given the model order, we estimate the component means by score-initialized gradient descent on a MUSIC-type projection objective. Under an explicit sample-size condition, a qualifying sample initialization lies in a certified attraction region with high probability, after which the iterates converge linearly. For fixed positive component separation, the resulting mean estimates achieve the parametric rate

\mathcal{O}_p(n^{-1/2})

. Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.

Xinyu Liu, Hai Zhang

Department of Mathematics, Hong Kong University of Science and Technology (HKUST) · HKUST-Shenzhen-Hong Kong Collaborative Innovation Research Institute

Gaussian Mixture Models Spectral Methods

Mar 16, 2026astro-ph.CO

Spectral Hierarchy of the Cosmic Web

We introduce a spectral hierarchy of cosmic-web classifications obtained by applying simple scale-weighting kernels to the density field before performing a standard eigenvalue-based web classification. This unifies and extends several widely used web definitions within a single framework: the familiar potential/tidal web (large-scale, nonlocal), a curvature-based web (more local, peak- and ridge-sensitive), and additional higher-derivative levels that progressively emphasize smaller-scale structure. Because the classification is built from second derivatives of the filtered field, successive hierarchy levels align naturally with operator families that appear in renormalised bias and effective descriptions of large-scale structure, providing an explicit bridge between cosmic-web environments and long- and short-range nonlocal bias ingredients. We quantify the information content of the hierarchy with a compact statistic: we map each cell to one of four ordered web types (void, sheet, filament, knot), construct a corresponding ``web contrast'' field, and measure its cross-correlation with halos from the AbacusSummit simulation suite on a coarse mesh with

ΔL\simeq 5.5\,h^{-1}\mathrm{Mpc}

. We find that the hierarchy retains significant tracer-relevant information from very large scales down to the mesh Nyquist limit, with the more local (curvature/higher-derivative) levels dominating toward nonlinear scales. This makes the spectral hierarchy a practical, interpretable conditioning basis for fast mock-galaxy production and field-level modelling, and a flexible tool for studying environment-dependent clustering and assembly bias.

Francisco-Shu Kitaura, Francesco Sinigaglia

Instituto de Astrofísica de Canarias, s/n, E-38205, La Laguna, Tenerife, Spain · Departamento de Astrofísica, Universidad de La Laguna, E-38206, La Laguna, Tenerife, Spain · Institute for Fundamental Physics of the Universe, Via Beirut 2, I-34151 Trieste, Italy +3

Hierarchical Classification Spectral Methods

Feb 26, 2026stat.ML

Regular Fourier Features for Nonstationary Gaussian Processes

Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations. Spectral methods address this challenge by exploiting the Fourier representation and treating the spectral density as a probability distribution suitable for Monte Carlo approximation. Although this probabilistic interpretation is valid for stationary processes, it is overly restrictive for the nonstationary case, where spectral densities are generally not probability measures. To avoid this limitation, we propose regular Fourier features for harmonizable processes with one-dimensional inputs. Our method discretizes the spectral representation directly, preserving the correlation structure among spectral weights without requiring probability assumptions. Assuming finite spectral support, this yields an efficient low-rank approximation that is positive semi-definite by construction and consistent under mild regularity conditions. When the spectral density is unknown, the framework also extends to kernel learning from data, which we explore as a proof of concept. We demonstrate the approximation on locally stationary and harmonizable mixture kernels, the latter with a complex-valued spectral density. As a feasibility study, we then apply the kernel-learning extension to real and synthetic data, where it matches competitive baselines.

Arsalan Jawaid, Abdullah Karatas, Jörg Seewig

Institute of Measurement and Sensor Technology, University of Kaiserslautern-Landau, Kaiserslautern, Germany · Independent Researcher, Mannheim, Germany

Fourier Feature Embeddings Gaussian Processes Low-Rank Approximation Kernel Methods Spectral Methods

Feb 19, 2026cs.SI

Simplify to Amplify: Achieving Information-Theoretic Bounds with Fewer Steps in Spectral Community Detection

We propose a streamlined spectral algorithm for community detection in the two-community stochastic block model (SBM) under constant edge density assumptions. By reducing algorithmic complexity through the elimination of non-essential preprocessing steps, our method directly leverages the spectral properties of the adjacency matrix. We demonstrate that our algorithm exploits specific characteristics of the second eigenvector to achieve improved error bounds that approach information-theoretic limits, representing a significant improvement over existing methods. Theoretical analysis establishes that our error rates are tighter than previously reported bounds in the literature. Comprehensive experimental validation confirms our theoretical findings and demonstrates the practical effectiveness of the simplified approach. Our results suggest that algorithmic simplification, rather than increasing complexity, can lead to both computational efficiency and enhanced performance in spectral community detection.

Sie Hendrata Dharmawan, Peter Chin

Thayer School Of Engineering · Dartmouth College · Hanover, NH, USA

Stochastic Block Model Graph Clustering Spectral Methods

Feb 10, 2026astro-ph.EP

Efficient reduction of stellar contamination and noise in planetary transmission spectra using neural networks

The characterization of exoplanetary atmospheres has been transformed by the James Webb Space Telescope (JWST), whose infrared sensitivity enables transmission spectroscopy at unprecedented precision. However, stellar heterogeneities (e.g., spots and faculae) remain a dominant source of contamination that can bias atmospheric retrievals if not properly corrected. We present a methodology for reducing stellar contamination and instrument-specific noise from exoplanet transmission spectra using neural networks, in particular the so-called Denoising AutoEncoders (DAEs). Our goals are to enable fast, accurate corrections that improve the reliability of atmospheric parameter retrievals and to promote the use of unsupervised algorithms for efficient data processing. We designed and trained DAE architectures using large synthetic datasets of terrestrial (TRAPPIST-1e analogues) and sub-Neptune (K2-18b analogues) planets. Atmospheric retrieval experiments were then performed on contaminated spectra in order to compare our deep-learning approach against standard correction methods in terms of accuracy and computational cost. Our autoencoders successfully reconstruct uncontaminated spectra, preserving essential molecular features even in low-S/N regimes. In retrieval tests, the denoising autoencoder pre-processing yields atmospheric parameter estimates broadly comparable to those obtained with simultaneous stellar-contamination fitting. Notably, our method maintains a much lower computational cost, approximately one order of magnitude smaller. These results demonstrate that DAEs outperform conventional correction methods in computational efficiency while maintaining high accuracy, paving the way for their integration into future atmospheric characterization pipelines for both rocky and sub-Neptune exoplanets.

David S. Duque-Castaño, Lauren Flor-Torres, Jorge I. Zuluaga

SEAP/FACom, Instituto de Física - FCEN, Universidad de Antioquia, Calle 70 No. 52-21, Medellín, Colombia

Denoising Autoencoders Autoencoders Spectral Methods Data Contamination

Feb 10, 2026cs.LG

Learning to Discover Iterative Spectral Algorithms

We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization. Our self-supervised models adapt to input operators using coarse spectral information (e.g., eigenvalue estimates and residual norms), and predict recurrence coefficients for computing or applying a matrix polynomial tailored to a downstream task. The effectiveness of AutoSpec relies on three ingredients: an architecture whose inference pass implements short, executable numerical linear algebra recurrences; efficient training on small synthetic problems with transfer to large-scale real-world operators; and task-defined objectives that enforce the desired approximation or preconditioning behavior across the range of spectral profiles represented in the training set. We apply AutoSpec to discovering algorithms for representative tasks on spd matrices: accelerating matrix function approximation; accelerating sparse linear solvers; and spectral filtering/preconditioning for eigenvalue computations. On real-world matrices, the learned procedures deliver up to order-of-magnitude improvements in accuracy and/or reductions in iteration count, relative to spectrum-agnostic baselines. We find clear connections to classical theory: the induced polynomials may exhibit equioscillation behavior characteristic of Chebyshev polynomial approximation. The code is available at: https://github.com/zihanghliu/AutoSpec .

Zihang Liu, Oleg Balabanov, Yaoqing Yang +1

1International Computer Science Institute · University of California, Berkeley · 4Lawrence Berkeley National Laboratory +1

Automated Algorithm Discovery Learning to Optimize Spectral Methods

Jan 29, 2026cs.CV

Correcting Spectra Outside the Backbone: A Model-Agnostic Rectifier for Hyperspectral Image Super-Resolution

Hyperspectral image super-resolution (HSI-SR) aims to recover spatial detail while preserving the spectral shape on which quantitative analysis relies. Recent HSI-SR methods, from repurposed RGB super-resolution backbones to dedicated spectral-spatial architectures, have greatly improved spatial reconstruction. However, overlooking the compact spectral structure of hyperspectral data leaves residual spectral errors, while binding the spectral treatment to each architecture forces it to be rebuilt for every new backbone. Yet the low-dimensional structure of spectra belongs to the data, not to any backbone. Backbones differ in the errors they leave, but not in the structure of the true spectra. One rectifier design can therefore serve any backbone. Building on this insight, we propose the \textbf{S}pectral \textbf{R}ectification \textbf{S}uper-\textbf{R}esolution Network (\textbf{SR

^{2}

-Net}), a model-agnostic rectifier that needs nothing from the backbone but its output, leaves its internal architecture untouched, and is trained per backbone. SR

^{2}

-Net follows an \emph{enhance-then-rectify} pipeline in which Hierarchical Spectral-Spatial Synergy Attention (\textbf{H-S

^{3}

A}) reinforces cross-band interactions, while Mode-Constrained Rectification (\textbf{MCR}) confines the correction to a learned compact spectral subspace. A degradation-consistency constraint further ties the output to the observed low-resolution input. Experiments with five backbones spanning CNN, Transformer, and diffusion families show that one fixed configuration improves spectral fidelity in every reported setting while preserving or improving spatial quality. Averaged over thirty in-domain settings, SR

^{2}

-Net removes 22.6% of the residual spectral error at a backbone-independent cost of 0.048M parameters.

Ji-Xuan He, Guohang Zhuang, Bo Junge +5

School of Computer Science and Technology, Xi’an Jiaotong University · School of Computer Science and Information Engineering, Hefei University of Technology

Hyperspectral Image Super-Resolution Hyperspectral Imaging Spectral Methods

Jan 23, 2026cs.LG

SFO: Learning PDE Operators via Spectral Filtering

Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.

Noam Koren, Rafael Moschopoulos, Kira Radinsky +1

Department of Computer Science Technion - Israel Institute of Technology · Department of Computer Science Princeton University

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Neural Operators

Dec 17, 2025physics.chem-ph

NMIRacle: Multi-modal Generative Molecular Elucidation from IR and NMR Spectra

Molecular structure elucidation from spectroscopic data is a long-standing challenge in Chemistry, traditionally requiring expert interpretation. We introduce NMIRacle, a two-stage generative framework that builds upon recent paradigms in AI-driven spectroscopy with minimal assumptions. In the first stage, NMIRacle learns to reconstruct molecular structures from count-aware fragment representations, capturing both fragment identities and their occurrences. In the second stage, a spectral encoder maps input spectra (IR, 1H-NMR, 13C-NMR) into a latent embedding used to condition the pre-trained generator, which is fine-tuned for direct spectra-to-molecule generation. This formulation bridges fragment-level chemical modeling with spectral evidence, yielding accurate molecular predictions. Empirical results demonstrate that NMIRacle outperforms existing baselines on molecular elucidation, while maintaining robust performance across increasing levels of molecular complexity.

Federico Ottomano, Yingzhen Li, Alex M. Ganose

Department of Chemistry, Imperial College London · Department of Computing, Imperial College London

Multimodal Generation Molecular Generation Spectral Methods Conditional Generative Modeling

Dec 9, 2025cs.LG

Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk network, which operates directly on raw spatial or spatiotemporal coordinates through fully connected layers, often struggles to represent sharp gradients, boundary layers, and other non-periodic solution structures on bounded domains. To address these limitations, we introduce the Spectral-Embedded Deep Operator Network (SEDONet), a novel DeepONet architecture in which the trunk is driven by a fixed Chebyshev spectral dictionary instead of coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias for bounded domains, enabling the learned operator to capture fine-scale features that are difficult for Fourier-based or MLP-only trunks to represent. SEDONet is evaluated on the 2-D Poisson equation, 1-D Burgers' equation, 1-D advection-diffusion equation, Allen-Cahn equation, Lorenz-96 chaotic system, and Darcy flow, covering elliptic, hyperbolic, parabolic, chaotic, and multiscale problems. Across all benchmarks, SEDONet consistently achieves the lowest or statistically comparable relative

L^2

errors among DeepONet, FEDONet, and SEDONet, with improvements of up to 54% over the baseline DeepONet and consistent gains over Fourier-embedded variants on bounded, non-periodic problems. Energy spectrum analyses further demonstrate that SEDONet more accurately preserves intermediate- and high-frequency solution structures. The proposed framework provides a simple, parameter-neutral modification to DeepONets, offering a robust and computationally efficient spectral approach for surrogate modeling of nonlinear operators in scientific computing.

Muhammad Abid, Omer San

Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Deep Operator Networks

Oct 2, 2025cs.LG

Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization

Estimating the tangent spaces of a data manifold is a fundamental problem in geometric data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise setting due to a critical trade-off in choosing the neighborhood size. Selecting an optimal size requires prior knowledge of the geometric and noise characteristics of the data that are often unavailable. In this paper, we propose a spectral method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation. Instead of relying solely on local neighborhoods, LEGO estimates the tangent space at each data point by orthogonalizing the gradients of low-frequency eigenvectors of the graph Laplacian. We provide two theoretical justifications of our method. First, a differential geometric analysis on the tubular neighborhood of a manifold shows that gradients of the low-frequency Neumann eigenfunctions of the tube align closely with the manifold's tangent bundle, while an eigenfunction with high gradient in directions orthogonal to the manifold lie deeper in the spectrum. Second, a random matrix theoretic analysis also demonstrates that low-frequency eigenvectors are robust to sub-Gaussian noise. These results allow us to derive the asymptotic scaling and stability of the estimated eigenvector gradients. Numerical experiments demonstrate that LEGO yields tangent space estimates that are significantly more robust to noise than those from LPCA, resulting in marked improvements in downstream tasks such as manifold learning, boundary detection, and local intrinsic dimension estimation.

Dhruv Kohli, Sawyer J. Robertson, Gal Mishne +1

Manifold Learning Spectral Methods

Feb 16, 2025quant-ph

Physics-Informed Support Vector Kernels via Green-Function Analogies and Jackson-Chebyshev Spectral Design

Kernel selection for regression of physical observables is often heuristic. We investigate a physics-informed strategy in which functional forms and spectral structures associated with Green's functions motivate kernel selection without requiring an exact identification between a machine-learning kernel and a physical propagator. The principal construction is a Jackson-damped Chebyshev kernel inspired by the kernel polynomial method (KPM); its explicit feature map yields a positive-semidefinite Gram matrix by construction and provides an inspectable spectral prior for structured observables. We evaluate standard and custom SVR models on copper-conductivity proxies, local Dirac-like band dispersion, quartic-oscillator energy levels, photonic-crystal transmission, and Fibonacci-chain transmission using repeated nested validation, learning curves, random-forest and multilayer-perceptron baselines, and low-rank Nyström tests where relevant. The framework is intended for finite-data regression of precomputed observables while boundary conditions remain part of the physical model that generates those observables.

Nan-Hong Kuo, Renata Wong

Department of Physics, National Taiwan University, Taipei, Taiwan · Department of Artificial Intelligence, Chang Gung University, Taoyuan, Taiwan · Department of Neurology, Chang Gung Memorial Hospital, Keelung, Taiwan +1

Support Vector Machines Kernel Methods Physics-Informed ML Spectral Methods

Jan 18, 2025stat.ML

Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer

The principal objective of this work is twofold within nonparametric regression settings: (1) to establish the minimax optimal convergence rates for fixed-bandwidth Gaussian kernel spectral algorithms when the true regression function resides in a Sobolev space, and (2) to apply Gaussian spectral algorithms for achieving robust and adaptive transfer learning under concept shift. While minimax optimality of misspecified spectral algorithms has been established, existing guarantees are typically restricted to the non-saturation regime. We demonstrate that the infinite smoothness of fixed-bandwidth Gaussian kernels provides universal robustness to model misspecification by showing that this kernel choice enables any spectral algorithm to attain minimax optimal rates, provided the regularization parameter decays exponentially. This result effectively decouples optimality from the algorithm's inherent qualification. Building on this, we then advocate Gaussian spectral algorithms as powerful components in a learning framework for robust and adaptive transfer. Specifically, we derive the adaptive convergence rate of the excess risk for this framework and show that the rates are optimal up to logarithmic factors. Our results also reveal the impact of the magnitude of the concept shift and the sample size on the generalization error.

Haotian Lin, Matthew Reimherr

Department of Statistics, The Pennsylvania State University.

Spectral Regularization Concept Drift Transfer Learning Kernel Methods Spectral Methods Statistical Learning Theory

Jul 8, 2024math.DS

Adversarial dynamical systems characterize when data-driven learning succeeds or fails

Many systems resist analytical modeling, making data-driven inference of dynamics important. Yet data-driven methods can fail to converge or generalize, leaving open a central question: When can system behavior be learned reliably from data, and when is such learning impossible? We answer this question using adversarial dynamical systems to identify the boundary between accessible and inaccessible regimes. In Koopman operator learning, a leading framework for representing nonlinear dynamics through linear spectral objects, we design optimal data-driven spectral algorithms with convergence and certification guarantees under conditions arising broadly in physical systems. This yields a convergence theory for Koopman-operator approximations and resolves a longstanding open problem in Koopman spectral analysis. Conversely, by constructing adversarial systems, we prove matching impossibility results: without these conditions, no single-sequence limiting procedure can guarantee learning, regardless of data quality. These results sharply characterize when data-driven spectral learning can succeed and when it must fail. We validate the framework on oscillators, chaotic fluid flows and Arctic sea ice concentration forecasting. In the latter, we uncover hidden modes of Arctic sea ice decline, deliver long-range forecasts with geographic error bounds, and outperform state-of-the-art dynamical and deep learning models at substantially lower computational cost, enabling real-time deployment on standard CPUs.

Matthew J. Colbrook, Igor Mezić, Alexei Stepanenko

DAMTP, University of Cambridge, Cambridge, CB3 0WA, UK. · University of California, Santa Barbara, CA 93106, USA.

Koopman Operator Learning Spectral Methods Statistical Learning Theory Chaotic Time-Series Prediction

Jan 16, 2024stat.ML

Semidefinite programming relaxations and debiasing for MAXCUT-based clustering

In this paper, we consider the problem of partitioning a small data sample of size

n

drawn from a mixture of

2

sub-gaussian distributions in

\mathbb{R}^p

. We consider semidefinite programming relaxations of an integer quadratic program that is formulated essentially as finding the maximum cut on a graph, where edge weights in the cut represent dissimilarity scores between two nodes based on their

p

features. We define the signal-to-noise ratio (SNR) as

s^2 := \min\{n p γ^2, Δ^2\}

, where

Δ^2 := p γ

denotes the

\ell_2^2

distance between the two cluster centers. Our contributions are twofold. First, we provide a unified framework for analyzing three computationally efficient algorithms: SDP1, BalancedSDP, and Spectral clustering, yielding universal polynomial-rate misclassification guarantees for all three algorithms. Moreover, our theory allows for partial recovery (success rate

< 100\%

) as long as

s^2

is lower bounded by a constant. Second, we prove that the misclassification errors for SDP1 and BalancedSDP decay exponentially with respect to the SNR

s^2

and the BalancedSDP requires no explicit debiasing when the two clusters have equal sizes. To our knowledge, this is the first time such results are obtained for semidefinite relaxations of MAX CUT in population clustering. We provide simulation evidence illuminating the theoretical predictions.

Shuheng Zhou

University of California, Riverside, CA 92521

Semidefinite Programming Unsupervised Clustering Graph Clustering Spectral Methods

Feb 8, 2019stat.ML

Robust Streaming PCA

We consider streaming principal component analysis when the stochastic data generating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on convergence of any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja's algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments on synthetic and real-world datasets.

Daniel Bienstock, Minchan Jeong, Apurv Shukla +1

Graduate School of AI KAIST

Principal Component Analysis Streaming Algorithms Spectral Methods Non-Stationary Time Series

Date pendingquant-ph

Spectral Geometry and Bosonic-Bloch Probes: Explorations in Quantum Learning

This paper studies how spectral geometry emerges in quantum learning models and how it can be diagnosed with physically grounded probes. In graph-regularized quantum networks, training reorganizes the output similarity graph, increases the effective spectral dimension Delta S = +0.23, and reshapes the Laplacian spectrum. Edge-resolved two-boson interference directly probes this restructuring: the bosonic enhancement Delta P_uv correlates with the Fiedler edge split |Delta v_2| (r = -0.50), linking learned spectral partitions to interference signatures. A phase diagram shows a nonmonotonic dependence of performance on coupling strength gamma and noise delta, with graph regularization improving fidelity only in a restricted regime; hardware experiments confirm the predicted interference behavior within shot-noise uncertainty. We also analyze a hybrid quantum autoencoder and introduce Bloch-space drift as a geometric diagnostic of its latent representation. With an unsupervised benign-data threshold, the model achieves high ranking performance (ROC-AUC about 0.99) and negligible false-negative rates. Absolute Bloch drift strongly discriminates anomalies (ROC-AUC at least about 0.9), while consecutive drift is near random (ROC-AUC about 0.5), showing that detection arises from persistent state-space displacement rather than local fluctuations. Through the geometry of reduced single-qubit states and associated quantum Fisher information, these results show that learning-induced spectral organization appears as measurable quantum-state structure, establishing a unified spectral-geometric framework for diagnosing quantum learning systems with bosonic and Bloch probes.

Santanu Ganguly, Xing Liang, Dimitrios Makris

Quantum AI Research Group, School of Computer Science and Mathematics, Kingston University London, Penrhyn Road, Kingston upon Thames, KT1 2EE, United Kingdom

Quantum Machine Learning Spectral Methods Quantum Neural Networks