Daily Papers

From the dynamics of a broad class of classical mean-field glass models one may obtain a quantum model with finite zero-temperature entropy, a quantum transition at zero temperature, and a time-reparametrization (quasi-)invariance in the dynamical equations for correlations. The low eigenvalue spectrum of the resulting quantum model is directly related to the structure and exploration of metastable states in the landscape of the original classical glass model. This mapping reveals deep connections between classical glasses and the properties of SYK-like models.

The prediction of material degradation is an important problem to solve in many industries. Environmental conditions, such as humidity and temperature, are important drivers of degradation processes, with corrosion being one of the most prominent ones. Quantum machine learning is a promising research field but suffers from well known deficits such as barren plateaus and measurement overheads. To address this problem, recent research has examined quantum reservoir computing to address time-series prediction tasks. Although a promising idea, developing circuits that are expressive enough while respecting the limited depths available on current devices is challenging. In classical reservoir computing, the onion echo state network model (ESN) [https://doi.org/10.1007/978-3-031-72359-9_9] was introduced to increase the interpretability of the representation structure of the embeddings. This onion ESN model utilizes a concatenation of smaller reservoirs that describe different time scales by covering different regions of the eigenvalue spectrum. Here, we use the same idea in the realm of quantum reservoir computing by simultaneously evolving smaller quantum reservoirs to better capture all the relevant time-scales while keeping the circuit depth small. We do this by modifying the rotation angles which we show alters the eigenvalues of the quantum evolution, but also note that modifying the number of mid-circuit measurements accomplishes the same goals of changing the long-term or short-term memory. This onion QRC outperforms a simple model and a single classical reservoir for predicting the degradation of aluminum alloys in different environmental conditions. By combining the onion QRC with an additional classical reservoir layer, the prediction accuracy is further improved.

Quantization is an effective method for reducing memory footprint and inference time of Neural Networks, e.g., for efficient inference in the cloud, especially at the edge. However, ultra low precision quantization could lead to significant degradation in model generalization. A promising method to address this is to perform mixed-precision quantization, where more sensitive layers are kept at higher precision. However, the search space for a mixed-precision quantization is exponential in the number of layers. Recent work has proposed HAWQ, a novel Hessian based framework, with the aim of reducing this exponential search space by using second-order information. While promising, this prior work has three major limitations: (i) HAWQV1 only uses the top Hessian eigenvalue as a measure of sensitivity and do not consider the rest of the Hessian spectrum; (ii) HAWQV1 approach only provides relative sensitivity of different layers and therefore requires a manual selection of the mixed-precision setting; and (iii) HAWQV1 does not consider mixed-precision activation quantization. Here, we present HAWQV2 which addresses these shortcomings. For (i), we perform a theoretical analysis showing that a better sensitivity metric is to compute the average of all of the Hessian eigenvalues. For (ii), we develop a Pareto frontier based method for selecting the exact bit precision of different layers without any manual selection. For (iii), we extend the Hessian analysis to mixed-precision activation quantization. We have found this to be very beneficial for object detection. We show that HAWQV2 achieves new state-of-the-art results for a wide range of tasks.

We derive a sharp criterion on the spectra of periodic discrete Schrödinger operators acting on connected periodic lattices: the measure of the spectrum goes to zero as the coupling constant goes to infinity if and only if there is no infinite connected path of degeneracies.

The reciprocal distance Laplacian matrix of a connected graph G is defined as RD^L(G)=RT(G)-RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RD^L(G) is a real symmetric matrix, we denote its eigenvalues as λ_1(RD^L(G))geq λ_2(RD^L(G))geq dots geq λ_n(RD^L(G)). The largest eigenvalue λ_1(RD^L(G)) of RD^L(G) is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of n as a reciprocal distance Laplacian eigenvalue of RD^L(G) is exactly one less than the number of components in the complement graph G of G. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with n vertices. Also, we show that the star graph S_n is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with n vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph K_n, K_n-e, the star S_n, the complete balanced bipartite graph K_{n{2},n{2}} and the complete split graph CS(n,α) are all determined from the RD^L-spectrum.

Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning.

This thesis addresses two persistent and closely related challenges in modern deep learning, reliability and efficiency, through a unified framework grounded in Spectral Geometry and Random Matrix Theory (RMT). As deep networks and large language models continue to scale, their internal behavior becomes increasingly opaque, leading to hallucinations, fragile generalization under distribution shift, and growing computational and energy demands. By analyzing the eigenvalue dynamics of hidden activations across layers and inputs, this work shows that spectral statistics provide a compact, stable, and interpretable lens on model behavior, capable of separating structured, causal representations from noise-dominated variability. Within this framework, the first contribution, EigenTrack, introduces a real-time method for detecting hallucinations and out-of-distribution behavior in large language and vision-language models. EigenTrack transforms streaming activations into spectral descriptors such as entropy, variance, and deviations from the Marchenko-Pastur baseline, and models their temporal evolution using lightweight recurrent classifiers, enabling early detection of reliability failures before they appear in model outputs while offering interpretable insight into representation dynamics. The second contribution, RMT-KD, presents a principled approach to compressing deep networks via random matrix theoretic knowledge distillation. By interpreting outlier eigenvalues in activation spectra as carriers of task-relevant information, RMT-KD progressively projects networks onto lower-dimensional subspaces through iterative self-distillation, yielding significantly more compact and energy-efficient models while preserving accuracy and dense, hardware-friendly structure.

We look at the eigenvalues of the Hessian of a loss function before and after training. The eigenvalue distribution is seen to be composed of two parts, the bulk which is concentrated around zero, and the edges which are scattered away from zero. We present empirical evidence for the bulk indicating how over-parametrized the system is, and for the edges that depend on the input data.

We point out that the Lyapunov exponent of the eigenstate places restrictions on the eigenvalue. Consequently, with regard to non-Hermitian systems, even without any symmetry, the non-conservative Hamiltonians can exhibit real spectra as long as Lyapunov exponents of eigenstates inhibit imaginary parts of eigenvalues. Our findings open up a new route to study non-Hermitian physics.

This paper studies sequence modeling for prediction tasks with long range dependencies. We propose a new formulation for state space models (SSMs) based on learning linear dynamical systems with the spectral filtering algorithm (Hazan et al. (2017)). This gives rise to a novel sequence prediction architecture we call a spectral state space model. Spectral state space models have two primary advantages. First, they have provable robustness properties as their performance depends on neither the spectrum of the underlying dynamics nor the dimensionality of the problem. Second, these models are constructed with fixed convolutional filters that do not require learning while still outperforming SSMs in both theory and practice. The resulting models are evaluated on synthetic dynamical systems and long-range prediction tasks of various modalities. These evaluations support the theoretical benefits of spectral filtering for tasks requiring very long range memory.

We provide evidence of quantum kernel advantage under noiseless simulation in binary insurance classification on MIMIC-CXR chest radiographs using quantum support vector machines (QSVM) with frozen embeddings from three medical foundation models (MedSigLIP-448, RAD-DINO, ViT-patch32). We propose a two-tier fair comparison framework in which both classifiers receive identical PCA-q features. At Tier 1 (untuned QSVM vs. untuned linear SVM, C = 1 both sides), QSVM wins minority-class F1 in all 18 tested configurations (17 at p < 0.001, 1 at p < 0.01). The classical linear kernel collapses to majority-class prediction on 90-100% of seeds at every qubit count, while QSVM maintains non-trivial recall. At q = 11 (MedSigLIP-448 plateau center), QSVM achieves mean F1 = 0.343 vs. classical F1 = 0.050 (F1 gain = +0.293, p < 0.001) without hyperparameter tuning. Under Tier 2 (untuned QSVM vs. C-tuned RBF SVM), QSVM wins all seven tested configurations (mean gain +0.068, max +0.112). Eigenspectrum analysis reveals quantum kernel effective rank reaches 69.80 at q = 11, far exceeding linear kernel rank, while classical collapse remains C-invariant. A full qubit sweep reveals architecture-dependent concentration onset across models. Code: https://github.com/sebasmos/qml-medimage

The recent surge in contrast-based graph self-supervised learning has prominently featured an intensified exploration of spectral cues. Spectral augmentation, which involves modifying a graph's spectral properties such as eigenvalues or eigenvectors, is widely believed to enhance model performance. However, an intriguing paradox emerges, as methods grounded in seemingly conflicting assumptions regarding the spectral domain demonstrate notable enhancements in learning performance. Through extensive empirical studies, we find that simple edge perturbations - random edge dropping for node-level and random edge adding for graph-level self-supervised learning - consistently yield comparable or superior performance while being significantly more computationally efficient. This suggests that the computational overhead of sophisticated spectral augmentations may not justify their practical benefits. Our theoretical analysis of the InfoNCE loss bounds for shallow GNNs further supports this observation. The proposed insights represent a significant leap forward in the field, potentially refining the understanding and implementation of graph self-supervised learning.

This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function Delta(lambda) . This can be used to determine the multiplicities of eigenvalues for concrete operators.

Large language models and deep neural networks achieve strong performance but suffer from reliability issues and high computational cost. This thesis proposes a unified framework based on spectral geometry and random matrix theory to address both problems by analyzing the eigenvalue structure of hidden activations. The first contribution, EigenTrack, is a real-time method for detecting hallucinations and out-of-distribution behavior in language and vision-language models using spectral features and their temporal dynamics. The second contribution, RMT-KD, is a principled compression method that identifies informative spectral components and applies iterative knowledge distillation to produce compact and efficient models while preserving accuracy. Together, these results show that spectral statistics provide interpretable and robust signals for monitoring uncertainty and guiding compression in large-scale neural networks.

The spectral behaviour of the synaptic matrix, representing the neuronal connection strengths, is an important tool to analyze the stability and transient dynamics of a typical brain as well as its learning process and memory capacity. The complexity of the brain due to large number of neurons as well as underlying transient mechanisms e.g. homeostasis, seizure or synaptic plasticity can lead to networks with time-varying degree and type of sparsity. This renders an exact determination of the synaptic matrix not only technically difficult but also meaningless, leaving its statistical analysis as the best available theoretical approach. This motivates us to pursue a spectral analysis of the synaptic matrix models with different type of sparsity and thereby analyze latter's role on various aspects of network dynamics and stability. Our results have potential relevance for detemining the type of synaptic sparsity required to induce a specific brain function or desired transient mechanism e.g for pharmacological effects or physiological modulators.

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

This paper introduces a novel method for eigenvalue computation using a distributed cooperative neural network framework. Unlike traditional techniques that face scalability challenges in large systems, our decentralized algorithm enables multiple autonomous agents to collaboratively estimate the smallest eigenvalue of large matrices. Each agent employs a localized neural network, refining its estimates through communication with neighboring agents. Our empirical results confirm the algorithm's convergence towards the true eigenvalue, with estimates clustered closely around the true value. Even in the presence of communication delays or network disruptions, the method demonstrates strong robustness and scalability. Theoretical analysis further validates the accuracy and stability of the proposed approach, while empirical tests highlight its efficiency and precision, surpassing traditional centralized algorithms in large-scale eigenvalue computations.

Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we propose a general method to solve the localization problem by learning a regularization matrix from the localized eigenvectors. Using matrix perturbation analysis, we demonstrate that the learned regularizations suppress down the eigenvalues associated with localized eigenvectors and enable us to recover the informative eigenvectors representing the global structure. We show applications of our method in several inference problems: community detection in networks, clustering from pairwise similarities, rank estimation and matrix completion problems. Using extensive experiments, we illustrate that our method solves the localization problem and works down to the theoretical detectability limits in different kinds of synthetic data. This is in contrast with existing spectral algorithms based on data matrix, non-backtracking matrix, Laplacians and those with rank-one regularizations, which perform poorly in the sparse case with noise.

We introduce a novel framework that directly learns a spectral basis for shape and manifold analysis from unstructured data, eliminating the need for traditional operator selection, discretization, and eigensolvers. Grounded in optimal-approximation theory, we train a network to decompose an implicit approximation operator by minimizing the reconstruction error in the learned basis over a chosen distribution of probe functions. For suitable distributions, they can be seen as an approximation of the Laplacian operator and its eigendecomposition, which are fundamental in geometry processing. Furthermore, our method recovers in a unified manner not only the spectral basis, but also the implicit metric's sampling density and the eigenvalues of the underlying operator. Notably, our unsupervised method makes no assumption on the data manifold, such as meshing or manifold dimensionality, allowing it to scale to arbitrary datasets of any dimension. On point clouds lying on surfaces in 3D and high-dimensional image manifolds, our approach yields meaningful spectral bases, that can resemble those of the Laplacian, without explicit construction of an operator. By replacing the traditional operator selection, construction, and eigendecomposition with a learning-based approach, our framework offers a principled, data-driven alternative to conventional pipelines. This opens new possibilities in geometry processing for unstructured data, particularly in high-dimensional spaces.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

For a real quadratic field Q(d), we study the norm-form energy N = S_ζ^2 - d cdot S_L^2, where S_ζ and S_L are Lorentzian-weighted zero sums with w(ρ) = 2/(1/4 + γ^2). We prove three main results. (1) Spacelike spectral data: N < 0 unconditionally for all squarefree d > 1, as a consequence of a low-lying zero dominance theorem proved via explicit zero-counting. (2) Effective density bound: at each verified truncation level M, dens{N > 0} leq 2|f_{S_L^{(M)}}|_infty cdot (W_1(ζ)/d + ε_M), established unconditionally via Jacobi--Anger resonance analysis. (3) Exact asymptotic: under the computationally verified hypothesis that the infinite resonance lattice Λ_infty has finite rank (verified for M leq 20, where rank = 0), the sharp asymptotic dens{N > 0} = C(d)/d + o(1/d) holds. For d = 5, C(5) = 2,f_{S_L}(0)cdotE[|S_ζ|] = 0.1191; the constant depends on d through the zeros of L(s,χ_d), and C(d) = O(1/log d) as d to infty.

We examine the geometry of neural network training using the Jacobian of trained network parameters with respect to their initial values. Our analysis reveals low-dimensional structure in the training process which is dependent on the input data but largely independent of the labels. We find that the singular value spectrum of the Jacobian matrix consists of three distinctive regions: a "chaotic" region of values orders of magnitude greater than one, a large "bulk" region of values extremely close to one, and a "stable" region of values less than one. Along each bulk direction, the left and right singular vectors are nearly identical, indicating that perturbations to the initialization are carried through training almost unchanged. These perturbations have virtually no effect on the network's output in-distribution, yet do have an effect far out-of-distribution. While the Jacobian applies only locally around a single initialization, we find substantial overlap in bulk subspaces for different random seeds. Our code is available at https://github.com/EleutherAI/training-jacobian

Molecular structure elucidation from spectra is a foundational problem in chemistry, with profound implications for compound identification, synthesis, and drug development. Traditional methods rely heavily on expert interpretation and lack scalability. Pioneering machine learning methods have introduced retrieval-based strategies, but their reliance on finite libraries limits generalization to novel molecules. Generative models offer a promising alternative, yet most adopt autoregressive SMILES-based architectures that overlook 3D geometry and struggle to integrate diverse spectral modalities. In this work, we present DiffSpectra, a generative framework that directly infers both 2D and 3D molecular structures from multi-modal spectral data using diffusion models. DiffSpectra formulates structure elucidation as a conditional generation process. Its denoising network is parameterized by Diffusion Molecule Transformer, an SE(3)-equivariant architecture that integrates topological and geometric information. Conditioning is provided by SpecFormer, a transformer-based spectral encoder that captures intra- and inter-spectral dependencies from multi-modal spectra. Extensive experiments demonstrate that DiffSpectra achieves high accuracy in structure elucidation, recovering exact structures with 16.01% top-1 accuracy and 96.86% top-20 accuracy through sampling. The model benefits significantly from 3D geometric modeling, SpecFormer pre-training, and multi-modal conditioning. These results highlight the effectiveness of spectrum-conditioned diffusion modeling in addressing the challenge of molecular structure elucidation. To our knowledge, DiffSpectra is the first framework to unify multi-modal spectral reasoning and joint 2D/3D generative modeling for de novo molecular structure elucidation.

Identifying a small molecule from its mass spectrum is the primary open problem in computational metabolomics. This is typically cast as information retrieval: an unknown spectrum is matched against spectra predicted computationally from a large database of chemical structures. However, current approaches to spectrum prediction model the output space in ways that force a tradeoff between capturing high resolution mass information and tractable learning. We resolve this tradeoff by casting spectrum prediction as a mapping from an input molecular graph to a probability distribution over molecular formulas. We discover that a large corpus of mass spectra can be closely approximated using a fixed vocabulary constituting only 2% of all observed formulas. This enables efficient spectrum prediction using an architecture similar to graph classification - GrAFF-MS - achieving significantly lower prediction error and orders-of-magnitude faster runtime than state-of-the-art methods.

Designing effective positional encodings for graphs is key to building powerful graph transformers and enhancing message-passing graph neural networks. Although widespread, using Laplacian eigenvectors as positional encodings faces two fundamental challenges: (1) Non-uniqueness: there are many different eigendecompositions of the same Laplacian, and (2) Instability: small perturbations to the Laplacian could result in completely different eigenspaces, leading to unpredictable changes in positional encoding. Despite many attempts to address non-uniqueness, most methods overlook stability, leading to poor generalization on unseen graph structures. We identify the cause of instability to be a "hard partition" of eigenspaces. Hence, we introduce Stable and Expressive Positional Encodings (SPE), an architecture for processing eigenvectors that uses eigenvalues to "softly partition" eigenspaces. SPE is the first architecture that is (1) provably stable, and (2) universally expressive for basis invariant functions whilst respecting all symmetries of eigenvectors. Besides guaranteed stability, we prove that SPE is at least as expressive as existing methods, and highly capable of counting graph structures. Finally, we evaluate the effectiveness of our method on molecular property prediction, and out-of-distribution generalization tasks, finding improved generalization compared to existing positional encoding methods.

The mobility edge (ME) is a critical energy delineates the boundary between extended and localized states within the energy spectrum, and it plays a crucial role in understanding the metal-insulator transition in disordered or quasiperiodic systems. While there have been extensive studies on MEs in one-dimensional non-Hermitian (NH) quasiperiodic lattices recently, the investigation of exact NH MEs in two-dimensional (2D) cases remains rare. In the present study, we introduce a 2D dissipative Lieb lattice (DLL) model with imaginary quasiperiodic potentials applied solely to the vertices of the Lieb lattice. By mapping this DLL model to the 2D NH Aubry-Andr{é}-Harper (AAH) model, we analytically derive the exact ME and find it associated with the absolute eigenenergies. We find that the eigenvalues of extended states are purely imaginary when the quasiperiodic potential is strong enough. Additionally, we demonstrate that the introduction of imaginary quasiperiodic potentials does not disrupt the flat bands inherent in the system. Finally, we propose a theoretical framework for realizing our model using the Lindblad master equation. Our results pave the way for further investigation of exact NH MEs and flat bands in 2D dissipative quasiperiodic systems.

Artificial intelligence has made great strides in the last decade but still falls short of the human brain, the best-known example of intelligence. Not much is known of the neural processes that allow the brain to make the leap to achieve so much from so little beyond its ability to create knowledge structures that can be flexibly and dynamically combined, recombined, and applied in new and novel ways. This paper proposes a mathematical approach using graph theory and spectral graph theory, to hypothesize how to constrain these neural clusters of information based on eigen-relationships. This same hypothesis is hierarchically applied to scale up from the smallest to the largest clusters of knowledge that eventually lead to model building and reasoning.

Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.

We develop a Hellmann--Feynman type perturbation theory for the discrete signed p-Laplacian and apply it to a parametrized perturbation by edge cuts. We show that the eigenvalues of the signed p-Laplacian can be characterized as citical values of the parameter-dependent eigenvalues of a simpler graph.

While the harmonic function solution performs well in many semi-supervised learning (SSL) tasks, it is known to scale poorly with the number of samples. Recent successful and scalable methods, such as the eigenfunction method focus on efficiently approximating the whole spectrum of the graph Laplacian constructed from the data. This is in contrast to various subsampling and quantization methods proposed in the past, which may fail in preserving the graph spectra. However, the impact of the approximation of the spectrum on the final generalization error is either unknown, or requires strong assumptions on the data. In this paper, we introduce Sparse-HFS, an efficient edge-sparsification algorithm for SSL. By constructing an edge-sparse and spectrally similar graph, we are able to leverage the approximation guarantees of spectral sparsification methods to bound the generalization error of Sparse-HFS. As a result, we obtain a theoretically-grounded approximation scheme for graph-based SSL that also empirically matches the performance of known large-scale methods.

Transformers can learn to perform numerical computations from examples only. I study nine problems of linear algebra, from basic matrix operations to eigenvalue decomposition and inversion, and introduce and discuss four encoding schemes to represent real numbers. On all problems, transformers trained on sets of random matrices achieve high accuracies (over 90%). The models are robust to noise, and can generalize out of their training distribution. In particular, models trained to predict Laplace-distributed eigenvalues generalize to different classes of matrices: Wigner matrices or matrices with positive eigenvalues. The reverse is not true.

Diagnosing deep neural networks (DNNs) through the eigenspectrum of weight matrices has been an active area of research in recent years. At a high level, eigenspectrum analysis of DNNs involves measuring the heavytailness of the empirical spectral densities (ESD) of weight matrices. It provides insight into how well a model is trained and can guide decisions on assigning better layer-wise training hyperparameters. In this paper, we address a challenge associated with such eigenspectrum methods: the impact of the aspect ratio of weight matrices on estimated heavytailness metrics. We demonstrate that matrices of varying sizes (and aspect ratios) introduce a non-negligible bias in estimating heavytailness metrics, leading to inaccurate model diagnosis and layer-wise hyperparameter assignment. To overcome this challenge, we propose FARMS (Fixed-Aspect-Ratio Matrix Subsampling), a method that normalizes the weight matrices by subsampling submatrices with a fixed aspect ratio. Instead of measuring the heavytailness of the original ESD, we measure the average ESD of these subsampled submatrices. We show that measuring the heavytailness of these submatrices with the fixed aspect ratio can effectively mitigate the aspect ratio bias. We validate our approach across various optimization techniques and application domains that involve eigenspectrum analysis of weights, including image classification in computer vision (CV) models, scientific machine learning (SciML) model training, and large language model (LLM) pruning. Our results show that despite its simplicity, FARMS uniformly improves the accuracy of eigenspectrum analysis while enabling more effective layer-wise hyperparameter assignment in these application domains. In one of the LLM pruning experiments, FARMS reduces the perplexity of the LLaMA-7B model by 17.3% when compared with the state-of-the-art method.

We present a new method for generating emission spectra from polycyclic aromatic hydrocarbons (PAHs) in arbitrary radiation fields. We utilize the single-photon limit for PAH heating and emission to treat individual photon absorptions as independent events. This allows the construction of a set of single-photon emission "basis spectra" that can be scaled to produce an output emission spectrum given any input heating spectrum. We find that this method produces agreement with PAH emission spectra computed accounting for multi-photon effects to within simeq10% in the 3-20~{rm mu m} wavelength range for radiation fields with intensity U<100. We use this framework to explore the dependence of PAH band ratios on the radiation field spectrum across grain sizes, finding in particular a strong dependence of the 3.3 to 11.2~mum band ratio on radiation field hardness. A Python-based tool and a set of basis spectra that can be used to generate these emission spectra are made publicly available.

The paper is a continuation of the study started in Yorzh1. Schrodinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of delta type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of L_1 and C^M edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.

We consider fermions in a zero-temperature superconducting anti-de Sitter domain wall solution and find continuous bands of normal modes. These bands can be either partially filled or totally empty and gapped. We present a semi-classical argument which approximately captures the main features of the normal mode spectrum.

The analysis of the spectral features of a Toeplitz matrix-sequence left{T_{n}(f)right}_{ninmathbb N}, generated by a symbol fin L^1([-π,π]), real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when f is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form \[ T_{n}=c_0T_{n}(f_0)+c_{1} h^h T_{n}(f_{1})+c_{2} h^{2h} T_{n}(f_{2})+\cdots+c_{n-1} h^{(n-1)h}T_{n}(f_{n-1}), \] c_0,c_{1},ldots, c_{n-1} in [c_*,c^*], c^*ge c_*>0, independent of n, h=1{n}, f_jsim g_j, g_j=|θ|^{2-jh}, j=0,ldots,n-1. Since the resulting generating function depends on n, the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed together with open questions and future investigations.

We study the properties of common loss surfaces through their Hessian matrix. In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk. We present numerical evidence and mathematical justifications to the following conjectures laid out by Sagun et al. (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers. We believe that our observations have striking implications for non-convex optimization in high dimensions. First, the flatness of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading. And that the discussion of wide/narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create large connected components at the bottom of the landscape. Second, the dependence of small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs. With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light into the geometry of high-dimensional and non-convex spaces in modern applications. In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.

Convolutional Neural Networks (CNN) have been successful in processing data signals that are uniformly sampled in the spatial domain (e.g., images). However, most data signals do not natively exist on a grid, and in the process of being sampled onto a uniform physical grid suffer significant aliasing error and information loss. Moreover, signals can exist in different topological structures as, for example, points, lines, surfaces and volumes. It has been challenging to analyze signals with mixed topologies (for example, point cloud with surface mesh). To this end, we develop mathematical formulations for Non-Uniform Fourier Transforms (NUFT) to directly, and optimally, sample nonuniform data signals of different topologies defined on a simplex mesh into the spectral domain with no spatial sampling error. The spectral transform is performed in the Euclidean space, which removes the translation ambiguity from works on the graph spectrum. Our representation has four distinct advantages: (1) the process causes no spatial sampling error during the initial sampling, (2) the generality of this approach provides a unified framework for using CNNs to analyze signals of mixed topologies, (3) it allows us to leverage state-of-the-art backbone CNN architectures for effective learning without having to design a particular architecture for a particular data structure in an ad-hoc fashion, and (4) the representation allows weighted meshes where each element has a different weight (i.e., texture) indicating local properties. We achieve results on par with the state-of-the-art for the 3D shape retrieval task, and a new state-of-the-art for the point cloud to surface reconstruction task.

Time-domain surveys such as the Zwicky Transient Facility (ZTF) have opened a new frontier in the discovery and characterization of transients. While photometric light curves provide broad temporal coverage, spectroscopic observations remain crucial for physical interpretation and source classification. However, existing spectral analysis methods -- often reliant on template fitting or parametric models -- are limited in their ability to capture the complex and evolving spectra characteristic of such sources, which are sometimes only available at low resolution. In this work, we introduce SpectraNet, a deep convolutional neural network designed to learn robust representations of optical spectra from transients. Our model combines multi-scale convolution kernels and multi-scale pooling to extract features from preprocessed spectra in a hierarchical and interpretable manner. We train and validate SpectraNet on low-resolution time-series spectra obtained from the Spectral Energy Distribution Machine (SEDM) and other instruments, demonstrating state-of-the-art performance in classification. Furthermore, in redshift prediction tasks, SpectraNet achieves a root mean squared relative redshift error of 0.02, highlighting its effectiveness in precise regression tasks as well.

Whitening loss offers a theoretical guarantee against feature collapse in self-supervised learning (SSL) with joint embedding architectures. Typically, it involves a hard whitening approach, transforming the embedding and applying loss to the whitened output. In this work, we introduce Spectral Transformation (ST), a framework to modulate the spectrum of embedding and to seek for functions beyond whitening that can avoid dimensional collapse. We show that whitening is a special instance of ST by definition, and our empirical investigations unveil other ST instances capable of preventing collapse. Additionally, we propose a novel ST instance named IterNorm with trace loss (INTL). Theoretical analysis confirms INTL's efficacy in preventing collapse and modulating the spectrum of embedding toward equal-eigenvalues during optimization. Our experiments on ImageNet classification and COCO object detection demonstrate INTL's potential in learning superior representations. The code is available at https://github.com/winci-ai/INTL.

Large Language Models (LLMs) have demonstrated remarkable performance across various tasks but remain prone to hallucinations. Detecting hallucinations is essential for safety-critical applications, and recent methods leverage attention map properties to this end, though their effectiveness remains limited. In this work, we investigate the spectral features of attention maps by interpreting them as adjacency matrices of graph structures. We propose the LapEigvals method, which utilises the top-k eigenvalues of the Laplacian matrix derived from the attention maps as an input to hallucination detection probes. Empirical evaluations demonstrate that our approach achieves state-of-the-art hallucination detection performance among attention-based methods. Extensive ablation studies further highlight the robustness and generalisation of LapEigvals, paving the way for future advancements in the hallucination detection domain.

Providing a model that achieves a strong predictive performance and is simultaneously interpretable by humans is one of the most difficult challenges in machine learning research due to the conflicting nature of these two objectives. To address this challenge, we propose a modification of the radial basis function neural network model by equipping its Gaussian kernel with a learnable precision matrix. We show that precious information is contained in the spectrum of the precision matrix that can be extracted once the training of the model is completed. In particular, the eigenvectors explain the directions of maximum sensitivity of the model revealing the active subspace and suggesting potential applications for supervised dimensionality reduction. At the same time, the eigenvectors highlight the relationship in terms of absolute variation between the input and the latent variables, thereby allowing us to extract a ranking of the input variables based on their importance to the prediction task enhancing the model interpretability. We conducted numerical experiments for regression, classification, and feature selection tasks, comparing our model against popular machine learning models, the state-of-the-art deep learning-based embedding feature selection techniques, and a transformer model for tabular data. Our results demonstrate that the proposed model does not only yield an attractive prediction performance compared to the competitors but also provides meaningful and interpretable results that potentially could assist the decision-making process in real-world applications. A PyTorch implementation of the model is available on GitHub at the following link. https://github.com/dannyzx/Gaussian-RBFNN

Gradient signals in LLM training are highly anisotropic: recurrent linguistic structure concentrates energy into a small set of dominant spectral directions, while context specific information resides in a long tail. We show that this spike tail separation persists throughout training, with the spike occupying only about 1.5% of directions yet dominating optimizer statistics. This dominance suppresses tail learning by contracting tail updates through second moment normalization and tightening the globally stable learning rate bound. Motivated by this analysis, we propose Spectra, a spike aware optimizer that suppresses the dominant low rank spike subspace without amplifying the noise sensitive spectral tail. Spectra tracks the spike subspace via cached, warm started power iteration and applies low rank spectral shaping with negligible overhead and substantially reduced optimizer state memory. On LLaMA3 8B trained on 50B tokens, Spectra reaches the same target loss 30% faster than AdamW, reduces per step end to end overhead by 0.7%, cuts optimizer state memory by 49.25%, and improves average downstream accuracy by 1.62%. Compared to Muon, Spectra is 5.1x faster in optimizer processing time, achieves a lower final loss, and improves average accuracy by 0.66%.

Spectra are a prevalent yet highly information-dense form of scientific imagery, presenting substantial challenges to multimodal large language models (MLLMs) due to their unstructured and domain-specific characteristics. Here we introduce SpecVQA, a professional scientific-image benchmark for evaluating multimodal models on scientific spectral understanding, covering 7 representative spectrum types with expert-annotated question-answer pairs. The aim comprises two aspects: spectra scientific QA evaluation and corresponding underlying task evaluation. SpecVQA contains 620 figures and 3100 QA pairs curated from peer-reviewed literature, targeting both direct information extraction and domain-specific reasoning. To effectively reduce token length while preserving essential curve characteristics, we propose a spectral data sampling and interpolation reconstruction approach. Ablation studies further confirm that the approach achieves substantial performance improvements on the proposed benchmark. We test the capability of prominent MLLMs in scientific spectral understanding on our benchmark and present a leaderboard. This work represents an essential step toward enhancing spectral understanding in multimodal large models and suggests promising directions for extending visual-language models to broader scientific research and data analysis.

Graph-level anomaly detection has gained significant attention as it finds applications in various domains, such as cancer diagnosis and enzyme prediction. However, existing methods fail to capture the spectral properties of graph anomalies, resulting in unexplainable framework design and unsatisfying performance. In this paper, we re-investigate the spectral differences between anomalous and normal graphs. Our main observation shows a significant disparity in the accumulated spectral energy between these two classes. Moreover, we prove that the accumulated spectral energy of the graph signal can be represented by its Rayleigh Quotient, indicating that the Rayleigh Quotient is a driving factor behind the anomalous properties of graphs. Motivated by this, we propose Rayleigh Quotient Graph Neural Network (RQGNN), the first spectral GNN that explores the inherent spectral features of anomalous graphs for graph-level anomaly detection. Specifically, we introduce a novel framework with two components: the Rayleigh Quotient learning component (RQL) and Chebyshev Wavelet GNN with RQ-pooling (CWGNN-RQ). RQL explicitly captures the Rayleigh Quotient of graphs and CWGNN-RQ implicitly explores the spectral space of graphs. Extensive experiments on 10 real-world datasets show that RQGNN outperforms the best rival by 6.74% in Macro-F1 score and 1.44% in AUC, demonstrating the effectiveness of our framework. Our code is available at https://github.com/xydong127/RQGNN.

Consider a topological surface Σ. We introduce the spectrum of a representation from the fundamental group of Σ to SL(2,R), which is a subset of projective measured lamination on the surface, which captures the directions along which the representation fails to be Fuchsian, and which characterizes the action of the mapping class group on this representation. In the case of the once-punctured torus, we show that the spectrum of a generic representation is a Cantor set, and that it completely describes the dynamics of the familly of locally constant cocycles above interval exchange transformations associated to the representation.

Several recent studies advocate the use of spectral discriminators, which evaluate the Fourier spectra of images for generative modeling. However, the effectiveness of the spectral discriminators is not well interpreted yet. We tackle this issue by examining the spectral discriminators in the context of perceptual image super-resolution (i.e., GAN-based SR), as SR image quality is susceptible to spectral changes. Our analyses reveal that the spectral discriminator indeed performs better than the ordinary (a.k.a. spatial) discriminator in identifying the differences in the high-frequency range; however, the spatial discriminator holds an advantage in the low-frequency range. Thus, we suggest that the spectral and spatial discriminators shall be used simultaneously. Moreover, we improve the spectral discriminators by first calculating the patch-wise Fourier spectrum and then aggregating the spectra by Transformer. We verify the effectiveness of the proposed method twofold. On the one hand, thanks to the additional spectral discriminator, our obtained SR images have their spectra better aligned to those of the real images, which leads to a better PD tradeoff. On the other hand, our ensembled discriminator predicts the perceptual quality more accurately, as evidenced in the no-reference image quality assessment task.

Spectrum of ^7Li and elastic scattering reaction ^4He(^3H, ^3H)^4He are studied using the unified description of the Gamow shell model in the coupled-channel formulation (GSMCC). The reaction channels are constructed using the cluster expansion with the two mass partitions [^4He + ^3H], [^6Li + n].

Training Large Language Models (LLMs) at ultra-low precision is critically impeded by instability rooted in the conflict between discrete quantization constraints and the intrinsic heavy-tailed spectral nature of linguistic data. By formalizing the connection between Zipfian statistics and random matrix theory, we prove that the power-law decay in the singular value spectra of embeddings is a fundamental requisite for semantic encoding. We derive theoretical bounds showing that uniform quantization introduces a noise floor that disproportionately truncates this spectral tail, which induces spectral flattening and a strictly provable increase in the stable rank of representations. Empirical validation across diverse architectures including GPT-2 and TinyLlama corroborates that this geometric degradation precipitates representational collapse. This work not only quantifies the spectral sensitivity of LLMs but also establishes spectral fidelity as a necessary condition for stable low-bit optimization.

We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the Wronskian of suitable solutions of the two underlying difference equations. Until now only the case of perturbations of the main diagonal was known. We extend the known results to arbitrary perturbations, allow any (half-)open and closed spectral intervals, simplify the proof, and establish the comparison theorem.

Let G be a graph with adjacency matrix A(G). We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where n^+(G) and n^-(G) denote the number of positive and negative eigenvalues of A(G), respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torgasev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph G, its line graph L(G) satisfies n^+(L(G)) le n^-(L(G)) + 1 and obtain partial results.

For a graph G with n vertices and m edges, Lin et al. Lin define the atom--bond sum-connectivity (ABS) matrix of G such that the (i,j)^{th} entry is \[ 1 - \frac{2{d_i + d_j}} \] if vertex v_i is adjacent to the vertex v_j, and 0 otherwise. In this article, we determine the characteristic polynomial of the ABS matrix for certain specific classes of graphs. Furthermore, we compute the ABS eigenvalues and the ABS energy for these classes.

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics. Code is available at https://github.com/thuml/Latent-Spectral-Models.

Motivated by topologically protected states in wave physics, we study localised eigenmodes in one-dimensional periodic media with defects. The Su-Schrieffer-Heeger model (the canonical example of a one-dimensional system with topologically protected localised defect states) is used to demonstrate the method. Our approach can be used to describe two broad classes of perturbations to periodic differential problems: those caused by inserting a finite-sized piece of arbitrary material and those caused by creating an interface between two different periodic media. The results presented here characterise the existence of localised eigenmodes in each case and, when they exist, determine their eigenfrequencies and provide concise analytic results that quantify the decay rate of these modes. These results are obtained using both high-frequency homogenisation and transfer matrix analysis, with good agreement between the two methods.

We consider the convergence of the ESD for non-Hermitian random band matrices with independent entries to the circular law, which is the uniform measure on the unit disk in the center of the complex plane. We assume that the bandwidth of the matrix scales like n^γ for some γin(0,1], where n is the matrix size, and the variance profile of the matrix is only assumed to be doubly stochastic with no additional assumption on its specific mixing properties. We prove that the circular law limit holds either (1) when γ>5{6} and the entries are independent Gaussians, (2) or when γ>8{9} and the entries are independent subgaussian random variables. This new threshold improves the previous threshold γ>32{33} which was only proven for block band matrices and periodic band matrices. After the initial version of this paper, the author further extended the range of circular law for much smaller values of γ in 2508.18143 and 2511.01744 when the variance profile has specific mixing properties, but not for an arbitrary doubly stochastic variance profile. Thus the main contribution of this paper is the circular law for a genuine power law bandwidth for any doubly stochastic variance profile. We also prove an extended form of product circular law with a growing number of matrices. Weak delocalization estimates on eigenvectors are also derived. The new technical input is new polynomial lower bounds on some intermediate small singular values, and this estimate does not depend on the specific structure of the variance profile beyond the fact that it is doubly stochastic.

We study sampling algorithms for β-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of β-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size N mixes in O(log(N)).

One technique to visualize the training of neural networks is to perform PCA on the parameters over the course of training and to project to the subspace spanned by the first few PCA components. In this paper we compare this technique to the PCA of a high dimensional random walk. We compute the eigenvalues and eigenvectors of the covariance of the trajectory and prove that in the long trajectory and high dimensional limit most of the variance is in the first few PCA components, and that the projection of the trajectory onto any subspace spanned by PCA components is a Lissajous curve. We generalize these results to a random walk with momentum and to an Ornstein-Uhlenbeck processes (i.e., a random walk in a quadratic potential) and show that in high dimensions the walk is not mean reverting, but will instead be trapped at a fixed distance from the minimum. We finally compare the distribution of PCA variances and the PCA projected training trajectories of a linear model trained on CIFAR-10 and ResNet-50-v2 trained on Imagenet and find that the distribution of PCA variances resembles a random walk with drift.

In this paper, we propose a speed-up approach for subclass discriminant analysis and formulate a novel efficient multi-view solution to it. The speed-up approach is developed based on graph embedding and spectral regression approaches that involve eigendecomposition of the corresponding Laplacian matrix and regression to its eigenvectors. We show that by exploiting the structure of the between-class Laplacian matrix, the eigendecomposition step can be substituted with a much faster process. Furthermore, we formulate a novel criterion for multi-view subclass discriminant analysis and show that an efficient solution for it can be obtained in a similar to the single-view manner. We evaluate the proposed methods on nine single-view and nine multi-view datasets and compare them with related existing approaches. Experimental results show that the proposed solutions achieve competitive performance, often outperforming the existing methods. At the same time, they significantly decrease the training time.

We introduce NerVE, a unified eigenspectral framework for understanding how feed-forward networks (FFNs) in large language models (LLMs) organize and regulate information flow in high-dimensional latent space. Despite FFNs dominating the parameter budget, their high-dimensional dynamics remain poorly understood. NerVE addresses this gap through lightweight, memory-efficient tracking of eigenspectrum dynamics via four complementary metrics: Spectral Entropy (dispersion), Participation Ratio (effective dimensionality), Eigenvalue Early Enrichment (top-heaviness), and Jensen-Shannon divergence (distributional shifts). Our key insight is that FFN nonlinearities reinject variance across eigenmodes, fundamentally governing latent dimension utilization, and that optimizer geometry strongly modulates the extent of this variance reinjection. We validate NerVE across model scales, and diverse architectural and optimizer configurations, each uniquely shaping FFN dynamics: normalization schemes controlling variance flow; FFN weight geometries constraining latent space; positional encoding and activation functions regulating information flow; and optimizer choices redistributing effective capacity across depth. Across these settings, NerVE consistently recovers stable spectral signatures that correlate with model's generalization ability and respond predictably to design choices, generalizing beyond transformer to MLP-Mixer architectures, providing actionable insights for architectural and optimizer choices beyond trial-and-error.

Adapting large-scale pre-trained generative models in a parameter-efficient manner is gaining traction. Traditional methods like low rank adaptation achieve parameter efficiency by imposing constraints but may not be optimal for tasks requiring high representation capacity. We propose a novel spectrum-aware adaptation framework for generative models. Our method adjusts both singular values and their basis vectors of pretrained weights. Using the Kronecker product and efficient Stiefel optimizers, we achieve parameter-efficient adaptation of orthogonal matrices. We introduce Spectral Orthogonal Decomposition Adaptation (SODA), which balances computational efficiency and representation capacity. Extensive evaluations on text-to-image diffusion models demonstrate SODA's effectiveness, offering a spectrum-aware alternative to existing fine-tuning methods.

Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes mathcal{O}(d^{,3/2}/epsilon) queries to the time-evolution operator of the system and mathcal{O}(d^{,3/2}) queries to a block-encoding of the perturbation, for d cooling steps and an epsilon-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.

Linearization of attention using various kernel approximation and kernel learning techniques has shown promise. Past methods used a subset of combinations of component functions and weight matrices within the random feature paradigm. We identify the need for a systematic comparison of different combinations of weight matrices and component functions for attention learning in Transformer. Hence, we introduce Spectraformer, a unified framework for approximating and learning the kernel function in the attention mechanism of the Transformer. Our empirical results demonstrate, for the first time, that a random feature-based approach can achieve performance comparable to top-performing sparse and low-rank methods on the challenging Long Range Arena benchmark. Thus, we establish a new state-of-the-art for random feature-based efficient Transformers. The framework also produces many variants that offer different advantages in accuracy, training time, and memory consumption. Our code is available at: https://github.com/cruiseresearchgroup/spectraformer .

While spectral-based optimizers like Muon operate directly on the spectrum of updates, standard adaptive methods such as AdamW do not account for the global spectral structure of weights and gradients, leaving them vulnerable to two empirical issues in large language model (LLM) training: (i) the optimizer updates can have large spectral norms, potentially destabilizing training and degrading generalization; (ii) stochastic gradient noise can exhibit sparse spectral spikes, with a few dominant singular values much larger than the rest. We propose SPECTRA, a general framework addressing these by (i) post-spectral clipping of updates to enforce spectral-norm constraints; (ii) optional pre-spectral clipping of gradients to suppress spectral noise spikes. We prove that post-clipping constitutes a Composite Frank-Wolfe method with spectral-norm constraints and weight regularization, recovering Frobenius and ell_{infty}-norm regularization with SGD-based and sign-based methods. We further analyze how pre-clipping mitigates sparse spectral spikes. We propose efficient soft spectral clipping via Newton-Schulz iterations, avoiding expensive SVD. Experiments on LLM pretraining show SPECTRA uniformly improves validation loss for various optimizers, including AdamW, Signum, and AdEMAMix, with the best-performing variants achieving state-of-the-art results. Models trained with SPECTRA exhibit smaller weight norms, confirming the link between spectral clipping and regularization.

Hyperspectral image (HSI) plays a vital role in various fields such as agriculture and environmental monitoring. However, due to the expensive acquisition cost, the number of hyperspectral images is limited, degenerating the performance of downstream tasks. Although some recent studies have attempted to employ diffusion models to synthesize HSIs, they still struggle with the scarcity of HSIs, affecting the reliability and diversity of the generated images. Some studies propose to incorporate multi-modal data to enhance spatial diversity, but the spectral fidelity cannot be ensured. In addition, existing HSI synthesis models are typically uncontrollable or only support single-condition control, limiting their ability to generate accurate and reliable HSIs. To alleviate these issues, we propose HSIGene, a novel HSI generation foundation model which is based on latent diffusion and supports multi-condition control, allowing for more precise and reliable HSI generation. To enhance the spatial diversity of the training data while preserving spectral fidelity, we propose a new data augmentation method based on spatial super-resolution, in which HSIs are upscaled first, and thus abundant training patches could be obtained by cropping the high-resolution HSIs. In addition, to improve the perceptual quality of the augmented data, we introduce a novel two-stage HSI super-resolution framework, which first applies RGB bands super-resolution and then utilizes our proposed Rectangular Guided Attention Network (RGAN) for guided HSI super-resolution. Experiments demonstrate that the proposed model is capable of generating a vast quantity of realistic HSIs for downstream tasks such as denoising and super-resolution. The code and models are available at https://github.com/LiPang/HSIGene.

The {K_{1,1}, K_{1,2},C_m: mgeq3}-factor of a graph is a spanning subgraph whose each component is an element of {K_{1,1}, K_{1,2},C_m: mgeq3}. In this paper, through the graph spectral methods, we establish the lower bound of the signless Laplacian spectral radius and the upper bound of the distance spectral radius to determine whether a graph admits a {K_2}-factor. We get a lower bound on the size (resp. the spectral radius) of G to guarantee that G contains a {K_{1,1}, K_{1,2},C_m: mgeq3}-factor. Then we determine an upper bound on the distance spectral radius of G to ensure that G has a {K_{1,1}, K_{1,2},C_m: mgeq3}-factor. Furthermore, by constructing extremal graphs, we show that the above all bounds are best possible.

Unsupervised semantic segmentation is a long-standing challenge in computer vision with great significance. Spectral clustering is a theoretically grounded solution to it where the spectral embeddings for pixels are computed to construct distinct clusters. Despite recent progress in enhancing spectral clustering with powerful pre-trained models, current approaches still suffer from inefficiencies in spectral decomposition and inflexibility in applying them to the test data. This work addresses these issues by casting spectral clustering as a parametric approach that employs neural network-based eigenfunctions to produce spectral embeddings. The outputs of the neural eigenfunctions are further restricted to discrete vectors that indicate clustering assignments directly. As a result, an end-to-end NN-based paradigm of spectral clustering emerges. In practice, the neural eigenfunctions are lightweight and take the features from pre-trained models as inputs, improving training efficiency and unleashing the potential of pre-trained models for dense prediction. We conduct extensive empirical studies to validate the effectiveness of our approach and observe significant performance gains over competitive baselines on Pascal Context, Cityscapes, and ADE20K benchmarks.

This paper makes a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which results in the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. A second important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schr\"odinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism.

Dynamical systems provide a comprehensive way to study complex and changing behaviors across various sciences. Many modern systems are too complicated to analyze directly or we do not have access to models, driving significant interest in learning methods. Koopman operators have emerged as a dominant approach because they allow the study of nonlinear dynamics using linear techniques by solving an infinite-dimensional spectral problem. However, current algorithms face challenges such as lack of convergence, hindering practical progress. This paper addresses a fundamental open question: When can we robustly learn the spectral properties of Koopman operators from trajectory data of dynamical systems, and when can we not? Understanding these boundaries is crucial for analysis, applications, and designing algorithms. We establish a foundational approach that combines computational analysis and ergodic theory, revealing the first fundamental barriers -- universal for any algorithm -- associated with system geometry and complexity, regardless of data quality and quantity. For instance, we demonstrate well-behaved smooth dynamical systems on tori where non-trivial eigenfunctions of the Koopman operator cannot be determined by any sequence of (even randomized) algorithms, even with unlimited training data. Additionally, we identify when learning is possible and introduce optimal algorithms with verification that overcome issues in standard methods. These results pave the way for a sharp classification theory of data-driven dynamical systems based on how many limits are needed to solve a problem. These limits characterize all previous methods, presenting a unified view. Our framework systematically determines when and how Koopman spectral properties can be learned.

The asymptotically precise estimation of the generalization of kernel methods has recently received attention due to the parallels between neural networks and their associated kernels. However, prior works derive such estimates for training by kernel ridge regression (KRR), whereas neural networks are typically trained with gradient descent (GD). In the present work, we consider the training of kernels with a family of spectral algorithms specified by profile h(lambda), and including KRR and GD as special cases. Then, we derive the generalization error as a functional of learning profile h(lambda) for two data models: high-dimensional Gaussian and low-dimensional translation-invariant model. Under power-law assumptions on the spectrum of the kernel and target, we use our framework to (i) give full loss asymptotics for both noisy and noiseless observations (ii) show that the loss localizes on certain spectral scales, giving a new perspective on the KRR saturation phenomenon (iii) conjecture, and demonstrate for the considered data models, the universality of the loss w.r.t. non-spectral details of the problem, but only in case of noisy observation.

We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.

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

In this paper, we study the diffusability (learnability) of variational autoencoders (VAE) in latent diffusion. First, we show that pixel-space diffusion trained with an MSE objective is inherently biased toward learning low and mid spatial frequencies, and that the power-law power spectral density (PSD) of natural images makes this bias perceptually beneficial. Motivated by this result, we propose the Spectrum Matching Hypothesis: latents with superior diffusability should (i) follow a flattened power-law PSD (Encoding Spectrum Matching, ESM) and (ii) preserve frequency-to-frequency semantic correspondence through the decoder (Decoding Spectrum Matching, DSM). In practice, we apply ESM by matching the PSD between images and latents, and DSM via shared spectral masking with frequency-aligned reconstruction. Importantly, Spectrum Matching provides a unified view that clarifies prior observations of over-noisy or over-smoothed latents, and interprets several recent methods as special cases (e.g., VA-VAE, EQ-VAE). Experiments suggest that Spectrum Matching yields superior diffusion generation on CelebA and ImageNet datasets, and outperforms prior approaches. Finally, we extend the spectral view to representation alignment (REPA): we show that the directional spectral energy of the target representation is crucial for REPA, and propose a DoG-based method to further improve the performance of REPA. Our code is available https://github.com/forever208/SpectrumMatching.

Within the graph learning community, conventional wisdom dictates that spectral convolutional networks may only be deployed on undirected graphs: Only there could the existence of a well-defined graph Fourier transform be guaranteed, so that information may be translated between spatial- and spectral domains. Here we show this traditional reliance on the graph Fourier transform to be superfluous and -- making use of certain advanced tools from complex analysis and spectral theory -- extend spectral convolutions to directed graphs. We provide a frequency-response interpretation of newly developed filters, investigate the influence of the basis used to express filters and discuss the interplay with characteristic operators on which networks are based. In order to thoroughly test the developed theory, we conduct experiments in real world settings, showcasing that directed spectral convolutional networks provide new state of the art results for heterophilic node classification on many datasets and -- as opposed to baselines -- may be rendered stable to resolution-scale varying topological perturbations.

Previously, methods for learning marginally stable linear dynamical systems either required the transition matrix to be symmetric or incurred regret bounds that scale polynomially with the system's hidden dimension. In this work, we introduce a novel method that overcomes this trade-off, achieving dimension-free regret despite the presence of asymmetric matrices and marginal stability. Our method combines spectral filtering with linear predictors and employs Chebyshev polynomials in the complex plane to construct a novel spectral filtering basis. This construction guarantees sublinear regret in an online learning framework, without relying on any statistical or generative assumptions. Specifically, we prove that as long as the transition matrix has eigenvalues with complex component bounded by 1/poly log T, then our method achieves regret O(T^{9/10}) when compared to the best linear dynamical predictor in hindsight.

Large-amplitude molecular motions which occur during isomerization can cause significant changes in electronic structure. These variations in electronic properties can be used to identify vibrationally-excited eigenstates which are localized along the potential energy surface. This work demonstrates that nuclear quadrupole hyperfine interactions can be used as a diagnostic marker of progress along the isomerization path in both the HC14N/H14NC and DC15N/D15NC chemical systems. Ab initio calculations at the CCSD(T)/cc-pCVQZ level indicate that the hyperfine interaction is extremely sensitive to the chemical bonding of the quadrupolar 14N nucleus and can therefore be used to determine in which potential well the vibrational wavefunction is localized. A natural bonding orbital analysis along the isomerization path further demonstrates that hyperfine interactions arise from the asphericity of the electron density at the quadrupolar nucleus. Using the CCSD(T) potential surface, the quadrupole coupling constants of highly-excited vibrational states are computed from a one-dimensional internal coordinate path Hamiltonian. The excellent agreement between ab initio calculations and recent measurements demonstrates that nuclear quadrupole hyperfine structure can be used as a diagnostic tool for characterizing localized HCN and HNC vibrational states.

In recent years, large language models (LLMs) have transformed natural language understanding through vast datasets and large-scale parameterization. Inspired by this success, we present SpecCLIP, a foundation model framework that extends LLM-inspired methodologies to stellar spectral analysis. Stellar spectra, akin to structured language, encode rich physical and chemical information about stars. By training foundation models on large-scale spectral datasets, our goal is to learn robust and informative embeddings that support diverse downstream applications. As a proof of concept, SpecCLIP involves pre-training on two spectral types--LAMOST low-resolution and Gaia XP--followed by contrastive alignment using the CLIP (Contrastive Language-Image Pre-training) framework, adapted to associate spectra from different instruments. This alignment is complemented by auxiliary decoders that preserve spectrum-specific information and enable translation (prediction) between spectral types, with the former achieved by maximizing mutual information between embeddings and input spectra. The result is a cross-spectrum framework enabling intrinsic calibration and flexible applications across instruments. We demonstrate that fine-tuning these models on moderate-sized labeled datasets improves adaptability to tasks such as stellar-parameter estimation and chemical-abundance determination. SpecCLIP also enhances the accuracy and precision of parameter estimates benchmarked against external survey data. Additionally, its similarity search and cross-spectrum prediction capabilities offer potential for anomaly detection. Our results suggest that contrastively trained foundation models enriched with spectrum-aware decoders can advance precision stellar spectroscopy.

In this paper, we propose SPECTRUM, a temporal-frequency synergistic model that unlocks the untapped potential of multi-domain representation learning for online handwriting verification (OHV). SPECTRUM comprises three core components: (1) a multi-scale interactor that finely combines temporal and frequency features through dual-modal sequence interaction and multi-scale aggregation, (2) a self-gated fusion module that dynamically integrates global temporal and frequency features via self-driven balancing. These two components work synergistically to achieve micro-to-macro spectral-temporal integration. (3) A multi-domain distance-based verifier then utilizes both temporal and frequency representations to improve discrimination between genuine and forged handwriting, surpassing conventional temporal-only approaches. Extensive experiments demonstrate SPECTRUM's superior performance over existing OHV methods, underscoring the effectiveness of temporal-frequency multi-domain learning. Furthermore, we reveal that incorporating multiple handwritten biometrics fundamentally enhances the discriminative power of handwriting representations and facilitates verification. These findings not only validate the efficacy of multi-domain learning in OHV but also pave the way for future research in multi-domain approaches across both feature and biometric domains. Code is publicly available at https://github.com/NiceRingNode/SPECTRUM.

Establishing the relationship between 3D structures and the energy states of molecular systems has proven to be a promising approach for learning 3D molecular representations. However, existing methods are limited to modeling the molecular energy states from classical mechanics. This limitation results in a significant oversight of quantum mechanical effects, such as quantized (discrete) energy level structures, which offer a more accurate estimation of molecular energy and can be experimentally measured through energy spectra. In this paper, we propose to utilize the energy spectra to enhance the pre-training of 3D molecular representations (MolSpectra), thereby infusing the knowledge of quantum mechanics into the molecular representations. Specifically, we propose SpecFormer, a multi-spectrum encoder for encoding molecular spectra via masked patch reconstruction. By further aligning outputs from the 3D encoder and spectrum encoder using a contrastive objective, we enhance the 3D encoder's understanding of molecules. Evaluations on public benchmarks reveal that our pre-trained representations surpass existing methods in predicting molecular properties and modeling dynamics.

Vision-Language Models (VLMs) inherit adversarial vulnerabilities of Large Language Models (LLMs), which are further exacerbated by their multimodal nature. Existing defenses, including adversarial training, input transformations, and heuristic detection, are computationally expensive, architecture-dependent, and fragile against adaptive attacks. We introduce EigenShield, an inference-time defense leveraging Random Matrix Theory to quantify adversarial disruptions in high-dimensional VLM representations. Unlike prior methods that rely on empirical heuristics, EigenShield employs the spiked covariance model to detect structured spectral deviations. Using a Robustness-based Nonconformity Score (RbNS) and quantile-based thresholding, it separates causal eigenvectors, which encode semantic information, from correlational eigenvectors that are susceptible to adversarial artifacts. By projecting embeddings onto the causal subspace, EigenShield filters adversarial noise without modifying model parameters or requiring adversarial training. This architecture-independent, attack-agnostic approach significantly reduces the attack success rate, establishing spectral analysis as a principled alternative to conventional defenses. Our results demonstrate that EigenShield consistently outperforms all existing defenses, including adversarial training, UNIGUARD, and CIDER.

Scaling laws have made language-model performance predictable from model size, data, and compute, but they typically treat the optimizer as a fixed training detail. We show that this assumption misses a fundamental axis of representation scaling: how effectively the optimizer converts added FFN width into utilized spectral capacity. Using eigenspectra of feed-forward network representations, measured through soft and hard spectral-ranks, we find that the same Transformer architecture realizes markedly different spectral scaling laws when trained with different optimizers. Holding architecture and width schedule fixed, AdamW exhibits weak hard-rank scaling (β=0.44) on rare-token (TAIL) representations where learning is known to be hardest, whereas Muon achieves linear scaling (β=1.02) in the same regimes, a 2.3times increase in the scaling exponent. This difference is not reducible to validation loss: AdamW configurations can match low-rank Dion variants in perplexity, under extended training, while exhibiting sharply different spectral geometry, demonstrating that matched loss does not imply matched representation structure. Hard--soft rank asymmetry further reveals that optimizers differ not only in how much capacity is realized, but also in how that capacity is structured across eigenmodes. To disentangle optimizer effects from architectural ones, we compare against architectural interventions (e.g., attention rank and positional encoding), and find that optimizer-induced spectral shifts often exceed the architectural effects. These results suggest optimization as a first-class axis of representation scaling, motivating optimizer--architecture co-design.

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy Re=20,000 lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

Let F be a graph, and let B_r(F) be the class of r-uniform Berge-F hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. Based on the relationship, we give a spectral asymptotic bound for B_{r}(C_3)-free linear r-uniform hypergraphs and upper bounds for the spectral radii of B_{r}(K_{2,t})-free or {B_{r}(K_{s,t}),B_{r}(C_{3})}-free linear r-uniform hypergraphs, where C_{3} and K_{s,t} are respectively the triangle and the complete bipartite graph with one part having s vertices and the other part having t vertices. Our work implies an upper bound for the number of edges of {B_{r}(K_{s,t}),B_{r}(C_{3})}-free linear r-uniform hypergraphs and extends some of the existing research on (spectral) extremal problems of hypergraphs.

Recent works have established that AI models introduce spectral artifacts into generated images and propose approaches for learning to capture them using labeled data. However, the significant differences in such artifacts among different generative models hinder these approaches from generalizing to generators not seen during training. In this work, we build upon the key idea that the spectral distribution of real images constitutes both an invariant and highly discriminative pattern for AI-generated image detection. To model this under a self-supervised setup, we employ masked spectral learning using the pretext task of frequency reconstruction. Since generated images constitute out-of-distribution samples for this model, we propose spectral reconstruction similarity to capture this divergence. Moreover, we introduce spectral context attention, which enables our approach to efficiently capture subtle spectral inconsistencies in images of any resolution. Our spectral AI-generated image detection approach (SPAI) achieves a 5.5% absolute improvement in AUC over the previous state-of-the-art across 13 recent generative approaches, while exhibiting robustness against common online perturbations. Code is available on https://mever-team.github.io/spai.

Spectral graph neural networks (GNNs) are highly effective in modeling graph signals, with their success in recommendation often attributed to low-pass filtering. However, recent studies highlight the importance of high-frequency signals. The role of low-frequency and high-frequency graph signals in recommendation remains unclear. This paper aims to bridge this gap by investigating the influence of graph signals on recommendation performance. We theoretically prove that the effects of low-frequency and high-frequency graph signals are equivalent in recommendation tasks, as both contribute by smoothing the similarities between user-item pairs. To leverage this insight, we propose a frequency signal scaler, a plug-and-play module that adjusts the graph signal filter function to fine-tune the smoothness between user-item pairs, making it compatible with any GNN model. Additionally, we identify and prove that graph embedding-based methods cannot fully capture the characteristics of graph signals. To address this limitation, a space flip method is introduced to restore the expressive power of graph embeddings. Remarkably, we demonstrate that either low-frequency or high-frequency graph signals alone are sufficient for effective recommendations. Extensive experiments on four public datasets validate the effectiveness of our proposed methods. Code is avaliable at https://github.com/mojosey/SimGCF.

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable "perturb-then-diagonalize" (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

The ability to learn good representations of states is essential for solving large reinforcement learning problems, where exploration, generalization, and transfer are particularly challenging. The Laplacian representation is a promising approach to address these problems by inducing intrinsic rewards for temporally-extended action discovery and reward shaping, and informative state encoding. To obtain the Laplacian representation one needs to compute the eigensystem of the graph Laplacian, which is often approximated through optimization objectives compatible with deep learning approaches. These approximations, however, depend on hyperparameters that are impossible to tune efficiently, converge to arbitrary rotations of the desired eigenvectors, and are unable to accurately recover the corresponding eigenvalues. In this paper we introduce a theoretically sound objective and corresponding optimization algorithm for approximating the Laplacian representation. Our approach naturally recovers both the true eigenvectors and eigenvalues while eliminating the hyperparameter dependence of previous approximations. We provide theoretical guarantees for our method and we show that those results translate empirically into robust learning across multiple environments.

scqubits is an open-source Python package for simulating and analyzing superconducting circuits. It provides convenient routines to obtain energy spectra of common superconducting qubits, such as the transmon, fluxonium, flux, cos(2ϕ) and the 0-π qubit. scqubits also features a number of options for visualizing the computed spectral data, including plots of energy levels as a function of external parameters, display of matrix elements of various operators as well as means to easily plot qubit wavefunctions. Many of these tools are not limited to single qubits, but extend to composite Hilbert spaces consisting of coupled superconducting qubits and harmonic (or weakly anharmonic) modes. The library provides an extensive suite of methods for estimating qubit coherence times due to a variety of commonly considered noise channels. While all functionality of scqubits can be accessed programatically, the package also implements GUI-like widgets that, with a few clicks can help users both create relevant Python objects, as well as explore their properties through various plots. When applicable, the library harnesses the computing power of multiple cores via multiprocessing. scqubits further exposes a direct interface to the Quantum Toolbox in Python (QuTiP) package, allowing the user to efficiently leverage QuTiP's proven capabilities for simulating time evolution.

Previous research has shown that the principal singular vectors of a pre-trained model's weight matrices capture critical knowledge. In contrast, those associated with small singular values may contain noise or less reliable information. As a result, the LoRA-based parameter-efficient fine-tuning (PEFT) approach, which does not constrain the use of the spectral space, may not be effective for tasks that demand high representation capacity. In this study, we enhance existing PEFT techniques by incorporating the spectral information of pre-trained weight matrices into the fine-tuning process. We investigate spectral adaptation strategies with a particular focus on the additive adjustment of top singular vectors. This is accomplished by applying singular value decomposition (SVD) to the pre-trained weight matrices and restricting the fine-tuning within the top spectral space. Extensive speaker verification experiments on VoxCeleb1 and CN-Celeb1 demonstrate enhanced tuning performance with the proposed approach. Code is released at https://github.com/lizhepolyu/SpectralFT.

When rows of an n times d matrix A are given in a stream, we study algorithms for approximating the top eigenvector of the matrix {A}^TA (equivalently, the top right singular vector of A). We consider worst case inputs A but assume that the rows are presented to the streaming algorithm in a uniformly random order. We show that when the gap parameter R = σ_1(A)^2/σ_2(A)^2 = Ω(1), then there is a randomized algorithm that uses O(h cdot d cdot polylog(d)) bits of space and outputs a unit vector v that has a correlation 1 - O(1/R) with the top eigenvector v_1. Here h denotes the number of heavy rows in the matrix, defined as the rows with Euclidean norm at least |{A}|_F/d cdot operatorname{polylog(d)}. We also provide a lower bound showing that any algorithm using O(hd/R) bits of space can obtain at most 1 - Ω(1/R^2) correlation with the top eigenvector. Thus, parameterizing the space complexity in terms of the number of heavy rows is necessary for high accuracy solutions. Our results improve upon the R = Ω(log n cdot log d) requirement in a recent work of Price and Xun (FOCS 2024). We note that the algorithm of Price and Xun works for arbitrary order streams whereas our algorithm requires a stronger assumption that the rows are presented in a uniformly random order. We additionally show that the gap requirements in their analysis can be brought down to R = Ω(log^2 d) for arbitrary order streams and R = Ω(log d) for random order streams. The requirement of R = Ω(log d) for random order streams is nearly tight for their analysis as we obtain a simple instance with R = Ω(log d/loglog d) for which their algorithm, with any fixed learning rate, cannot output a vector approximating the top eigenvector v_1.

We present the Evolving Graph Fourier Transform (EFT), the first invertible spectral transform that captures evolving representations on temporal graphs. We motivate our work by the inadequacy of existing methods for capturing the evolving graph spectra, which are also computationally expensive due to the temporal aspect along with the graph vertex domain. We view the problem as an optimization over the Laplacian of the continuous time dynamic graph. Additionally, we propose pseudo-spectrum relaxations that decompose the transformation process, making it highly computationally efficient. The EFT method adeptly captures the evolving graph's structural and positional properties, making it effective for downstream tasks on evolving graphs. Hence, as a reference implementation, we develop a simple neural model induced with EFT for capturing evolving graph spectra. We empirically validate our theoretical findings on a number of large-scale and standard temporal graph benchmarks and demonstrate that our model achieves state-of-the-art performance.

We present Topological Point Cloud Clustering (TPCC), a new method to cluster points in an arbitrary point cloud based on their contribution to global topological features. TPCC synthesizes desirable features from spectral clustering and topological data analysis and is based on considering the spectral properties of a simplicial complex associated to the considered point cloud. As it is based on considering sparse eigenvector computations, TPCC is similarly easy to interpret and implement as spectral clustering. However, by focusing not just on a single matrix associated to a graph created from the point cloud data, but on a whole set of Hodge-Laplacians associated to an appropriately constructed simplicial complex, we can leverage a far richer set of topological features to characterize the data points within the point cloud and benefit from the relative robustness of topological techniques against noise. We test the performance of TPCC on both synthetic and real-world data and compare it with classical spectral clustering.

We investigate the exceptional points (EPs) and their pseudospectra in black hole perturbation theory. By considering a Gaussian bump modification to the Regge-Wheeler potential with variable amplitude, position, and width parameters, (varepsilon,d,σ_0), a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. We find that the vorticity ν=pm1/2 and the Berry phase γ=π for loops encircling the EL, while ν=0 and γ=0 for those do not encircle the EL. Through matrix perturbation theory, we prove that the ε-pseudospectrum contour size scales as ε^{1/q} at an EP, where q is the order of the largest Jordan block of the Hamiltonian-like operator, contrasting with the linear ε scaling at non-EPs. Numerical implements confirm this observation, demonstrating enhanced spectral instability at EPs for non-Hermitian systems including black holes.

This paper studies the problem of recovering cameras from a set of fundamental matrices. A set of fundamental matrices is said to be compatible if a set of cameras exists for which they are the fundamental matrices. We focus on the complete graph, where fundamental matrices for each pair of cameras are given. Previous work has established necessary and sufficient conditions for compatibility as rank and eigenvalue conditions on the n-view fundamental matrix obtained by concatenating the individual fundamental matrices. In this work, we show that the eigenvalue condition is redundant. We provide explicit homogeneous polynomials that describe necessary and sufficient conditions for compatibility in terms of the fundamental matrices and their epipoles. In this direction, we find that quadruple-wise compatibility is enough to ensure global compatibility for any number of cameras. We demonstrate that for four cameras, compatibility is generically described by triple-wise conditions and one additional equation involving all fundamental matrices.

As large language models (LLMs) become increasingly deployed across various industries, concerns regarding their reliability, particularly due to hallucinations-outputs that are factually inaccurate or irrelevant to user input-have grown. Our research investigates the relationship between the training process and the emergence of hallucinations to address a key gap in existing research that focuses primarily on post hoc detection and mitigation strategies. Using models from the Pythia suite (70M-12B parameters) and several hallucination detection metrics, we analyze hallucination trends throughout training and explore LLM internal dynamics. We introduce SEnsitive Neuron Dropout (SeND), a novel training protocol designed to mitigate hallucinations by reducing variance during training. SeND achieves this by deterministically dropping neurons with significant variability on a dataset, referred to as Sensitive Neurons. In addition, we develop an unsupervised hallucination detection metric, Efficient EigenScore (EES), which approximates the traditional EigenScore in 2x speed. This efficient metric is integrated into our protocol, allowing SeND to be both computationally scalable and effective at reducing hallucinations. Our empirical evaluation demonstrates that our approach improves LLM reliability at test time by up to 40% compared to normal training while also providing an efficient method to improve factual accuracy when adapting LLMs to domains such as Wikipedia and Medical datasets.

We rigorously study the joint evolution of training dynamics via stochastic gradient descent (SGD) and the spectra of empirical Hessian and gradient matrices. We prove that in two canonical classification tasks for multi-class high-dimensional mixtures and either 1 or 2-layer neural networks, the SGD trajectory rapidly aligns with emerging low-rank outlier eigenspaces of the Hessian and gradient matrices. Moreover, in multi-layer settings this alignment occurs per layer, with the final layer's outlier eigenspace evolving over the course of training, and exhibiting rank deficiency when the SGD converges to sub-optimal classifiers. This establishes some of the rich predictions that have arisen from extensive numerical studies in the last decade about the spectra of Hessian and information matrices over the course of training in overparametrized networks.

We investigate spectral features of bottomonium at high temperature, in particular the thermal mass shift and width of ground state S-wave and P-wave state. We employ and compare a range of methods for determining these features from lattice NRQCD correlators, including direct correlator analyses (multi-exponential fits and moments of spectral functions), linear methods (Backus-Gilbert, Tikhonov and HLT methods), and Bayesian methods for spectral function reconstruction (MEM and BR). We comment on the reliability and limitations of the various methods.

Periodic signals play an important role in daily lives. Although conventional sequential models have shown remarkable success in various fields, they still come short in modeling periodicity; they either collapse, diverge or ignore details. In this paper, we introduce a novel framework inspired by Fourier series to generate periodic signals. We first decompose the given signals into multiple sines and cosines and then conditionally generate periodic signals with the output components. We have shown our model efficacy on three tasks: reconstruction, imputation and conditional generation. Our model outperforms baselines in all tasks and shows more stable and refined results.

Recently, deep clustering methods have gained momentum because of the high representational power of deep neural networks (DNNs) such as autoencoder. The key idea is that representation learning and clustering can reinforce each other: Good representations lead to good clustering while good clustering provides good supervisory signals to representation learning. Critical questions include: 1) How to optimize representation learning and clustering? 2) Should the reconstruction loss of autoencoder be considered always? In this paper, we propose DEKM (for Deep Embedded K-Means) to answer these two questions. Since the embedding space generated by autoencoder may have no obvious cluster structures, we propose to further transform the embedding space to a new space that reveals the cluster-structure information. This is achieved by an orthonormal transformation matrix, which contains the eigenvectors of the within-class scatter matrix of K-means. The eigenvalues indicate the importance of the eigenvectors' contributions to the cluster-structure information in the new space. Our goal is to increase the cluster-structure information. To this end, we discard the decoder and propose a greedy method to optimize the representation. Representation learning and clustering are alternately optimized by DEKM. Experimental results on the real-world datasets demonstrate that DEKM achieves state-of-the-art performance.

Data balancing across multiple modalities and sources appears in various forms in foundation models in machine learning and AI, e.g. in CLIP and DINO. We show that data balancing across modalities and sources actually offers an unsuspected benefit: variance reduction. We present a non-asymptotic statistical bound that quantifies this variance reduction effect and relates it to the eigenvalue decay of Markov operators. Furthermore, we describe how various forms of data balancing in contrastive multimodal learning and self-supervised clustering can be better understood, and even improved upon, owing to our variance reduction viewpoint.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

We report an optical method of generating arbitrary polarization states by manipulating the thicknesses of a pair of uniaxial birefringent plates, the optical axes of which are set at a crossing angle of {\pi}/4. The method has the remarkable feature of being able to generate a distribution of arbitrary polarization states in a group of highly discrete spectra without spatially separating the individual spectral components. The target polarization-state distribution is obtained as an optimal solution through an exploration. Within a realistic exploration range, a sufficient number of near-optimal solutions are found. This property is also reproduced well by a concise model based on a distribution of exploration points on a Poincar\'e sphere, showing that the number of near-optimal solutions behaves according to a power law with respect to the number of spectral components of concern. As a typical example of an application, by applying this method to a set of phase-locked highly discrete spectra, we numerically demonstrate the continuous generation of a vector-like optical electric field waveform, the helicity of which is alternated within a single optical cycle in the time domain.

Deep neural networks are known to exhibit a spectral learning bias, wherein low-frequency components are learned early in training, while high-frequency modes emerge more gradually in later epochs. However, when the target signal lacks low-frequency components and is dominated by broadband high frequencies, training suffers from a 'spectral bottleneck', and the model fails to reconstruct the entire signal, including the frequency components that lie within the network's representational capacity. We examine such a scenario in the context of implicit neural representations (INRs) with sinusoidal representation networks (SIRENs), focusing on the challenge of fitting high-frequency-dominant signals that are susceptible to spectral bottleneck. To effectively fit any target signal irrespective of it's frequency content, we propose a generalized target-aware 'weight perturbation scheme' (WINNER - weight initialization with noise for neural representations) for network initialization. The scheme perturbs uniformly initialized weights with Gaussian noise, where the noise scales are adaptively determined by the spectral centroid of the target signal. We show that the noise scales can provide control over the spectra of network activations and the eigenbasis of the empirical neural tangent kernel. This method not only addresses the spectral bottleneck but also yields faster convergence and with improved representation accuracy, outperforming state-of-the-art approaches in audio fitting and achieving notable gains in image fitting and denoising tasks. Beyond signal reconstruction, our approach opens new directions for adaptive weight initialization strategies in computer vision and scientific machine learning.

One of the challenges in the study of generative adversarial networks is the instability of its training. In this paper, we propose a novel weight normalization technique called spectral normalization to stabilize the training of the discriminator. Our new normalization technique is computationally light and easy to incorporate into existing implementations. We tested the efficacy of spectral normalization on CIFAR10, STL-10, and ILSVRC2012 dataset, and we experimentally confirmed that spectrally normalized GANs (SN-GANs) is capable of generating images of better or equal quality relative to the previous training stabilization techniques.

Spectral analysis provides one of the most effective paradigms for information-preserving dimensionality reduction, as simple descriptions of naturally occurring signals are often obtained via few terms of periodic basis functions. In this work, we study deep neural networks designed to harness the structure in frequency domain for efficient learning of long-range correlations in space or time: frequency-domain models (FDMs). Existing FDMs are based on complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform computation on the spectrum and input data separately. This design introduces considerable computational overhead: for each layer, a forward and inverse FT. Instead, this work introduces a blueprint for frequency domain learning through a single transform: transform once (T1). To enable efficient, direct learning in the frequency domain we derive a variance-preserving weight initialization scheme and investigate methods for frequency selection in reduced-order FDMs. Our results noticeably streamline the design process of FDMs, pruning redundant transforms, and leading to speedups of 3x to 10x that increase with data resolution and model size. We perform extensive experiments on learning the solution operator of spatio-temporal dynamics, including incompressible Navier-Stokes, turbulent flows around airfoils and high-resolution video of smoke. T1 models improve on the test performance of FDMs while requiring significantly less computation (5 hours instead of 32 for our large-scale experiment), with over 20% reduction in average predictive error across tasks.

Accurate determination of qubit parameters is critical for the successful implementation of quantum information and computation applications. In solid state systems, the parameters of individual qubits vary across the entire system, requiring time consuming measurements and manual fitting processes for characterization. Recent developed superconducting qubits, such as fluxonium or 0-pi qubits, offer improved fidelity operations but exhibit a more complex physical and spectral structure, complicating parameter extraction. In this work, we propose a machine learning (ML)based methodology for the automatic and accurate characterization of fluxonium qubit parameters. Our approach utilized the energy spectrum calculated by a model Hamiltonian with various magnetic fields, as training data for the ML model. The output consists of the essential fluxonium qubit energy parameters, EJ, EC, and EL in Hamiltonian. The ML model achieves remarkable accuracy (with an average accuracy 95.6%) as an initial guess, enabling the development of an automatic fitting procedure for direct application to realistic experimental data. Moreover, we demonstrate that similar accuracy can be retrieved even when the input experimental spectrum is noisy or incomplete, highlighting the model robustness. These results suggest that our automated characterization method, based on a transfer learning approach, provides a reliable framework for future extensions to other superconducting qubits or different solid-state systems. Ultimately, we believe this methodology paves the way for the construction of large-scale quantum processors.

Learning PDE dynamics with neural solvers can significantly improve wall-clock efficiency and accuracy compared with classical numerical solvers. In recent years, foundation models for PDEs have largely adopted multi-scale windowed self-attention, with the scOT backbone in Poseidon serving as a representative example. However, because of their locality, truly globally consistent spectral coupling can only be propagated gradually through deep stacking and window shifting. This weakens global coupling and leads to error accumulation and drift during closed-loop rollouts. To address this, we propose DRIFT-Net. It employs a dual-branch design comprising a spectral branch and an image branch. The spectral branch is responsible for capturing global, large-scale low-frequency information, whereas the image branch focuses on local details and nonstationary structures. Specifically, we first perform controlled, lightweight mixing within the low-frequency range. Then we fuse the spectral and image paths at each layer via bandwise weighting, which avoids the width inflation and training instability caused by naive concatenation. The fused result is transformed back into the spatial domain and added to the image branch, thereby preserving both global structure and high-frequency details across scales. Compared with strong attention-based baselines, DRIFT-Net achieves lower error and higher throughput with fewer parameters under identical training settings and budget. On Navier--Stokes benchmarks, the relative L_{1} error is reduced by 7\%--54\%, the parameter count decreases by about 15\%, and the throughput remains higher than scOT. Ablation studies and theoretical analyses further demonstrate the stability and effectiveness of this design. The code is available at https://github.com/cruiseresearchgroup/DRIFT-Net.

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.

We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter Ntimes N random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of on-site random energies {a_i} and a structurally disordered hopping, we found that each eigenstate is delocalised over N^{2-gamma} sites close in energy |a_j-a_i|leq N^{1-gamma} in agreement with Kravtsov et al, arXiv:1508.01714. Our other main result, obtained combining a recurrence relation for the resolvent matrix with insights from Dyson's Brownian motion, is to show that the properties of the non-ergodic delocalised phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit.

As large language models (LLMs) scale, the question is not only how large they become, but how much of their capacity is effectively utilized. Existing scaling laws relate model size to loss, yet overlook how components exploit their latent space. We study feed-forward networks (FFNs) and recast width selection as a spectral utilization problem. Using a lightweight diagnostic suite -- Hard Rank (participation ratio), Soft Rank (Shannon rank), Spectral Concentration, and the composite Spectral Utilization Index (SUI) -- we quantify how many latent directions are meaningfully activated across LLaMA, GPT-2, and nGPT families. Our key finding is an asymmetric spectral scaling law: soft rank follows an almost perfect power law with FFN width, while hard rank grows only sublinearly and with high variance. This asymmetry suggests that widening FFNs mostly adds low-energy tail directions, while dominant-mode subspaces saturate early. Moreover, at larger widths, variance further collapses into a narrow subspace, leaving much of the latent space under-utilized. These results recast FFN width selection as a principled trade-off between tail capacity and dominant-mode capacity, offering concrete guidance for inference-efficient LLM design.

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

Spectral clustering is a popular clustering method. It first maps data into the spectral embedding space and then uses Kmeans to find clusters. However, the two decoupled steps prohibit joint optimization for the optimal solution. In addition, it needs to construct the similarity graph for samples, which suffers from the curse of dimensionality when the data are high-dimensional. To address these two challenges, we introduce Deep Spectral Clustering (DSC), which consists of two main modules: the spectral embedding module and the greedy Kmeans module. The former module learns to efficiently embed raw samples into the spectral embedding space using deep neural networks and power iteration. The latter module improves the cluster structures of Kmeans on the learned spectral embeddings by a greedy optimization strategy, which iteratively reveals the direction of the worst cluster structures and optimizes embeddings in this direction. To jointly optimize spectral embeddings and clustering, we seamlessly integrate the two modules and optimize them in an end-to-end manner. Experimental results on seven real-world datasets demonstrate that DSC achieves state-of-the-art clustering performance.

Auger electron spectroscopy, a way of characterizing electronic structure through core-level decay processes, is widely used in materials characterization; however direct calculation from molecular geometry requires accurate treatment of many excited states, posing a challenge for classical methods. We present a hybrid quantum-classical workflow for calculating Auger spectra that combines the generative quantum eigensolver (GQE) for ground-state preparation, the quantum self-consistent equation-of-motion method for excited-state calculations, and the one-centre approximation for Auger transition rates. GQE uses a GPT-2 model to generate quantum circuits for ground-state optimization, allowing our workflow to benefit from HPC parallelization and GPU-acceleration for favourable scaling with system size. We demonstrate the validity of our workflow by calculating the Auger spectrum of water with the STO-3G basis set and demonstrating qualitative and quantitative agreement with spectra obtained using completely classical full configuration interaction calculations, from the computational literature, and from the experimental literature. We also find that for water, substituting the variational quantum eigensolver (VQE) for GQE results in near-identical spectra, but that the ground state estimator generated by GQE contains about half the total gate count as that generated by VQE.

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: given an M-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 S_M requires exponential time in M. 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(d^2M^2 + d^3), where d 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) recovers a generative subgroup or only a supergroup.

We propose a novel spectral generative modeling framework for natural language processing that jointly learns a global time varying Fourier dictionary and per token mixing coefficients, replacing the ubiquitous self attention mechanism in transformer architectures. By enforcing reconstruction losses in both the time domain (embedding reconstruction) and the frequency domain (via Short Time Fourier Transform magnitude matching) alongside a standard language modeling objective, and fitting a Gaussian Mixture Model (GMM) prior over the learned mixing vectors, our approach achieves competitive perplexity and generation quality on standard benchmarks such as WikiText2 and Penn Treebank. In contrast to the quadratic computation complexity of self attention, our method operates with linear complexity, delivering substantial efficiency gains. We demonstrate that spectral dictionary models can achieve competitive performance compared to transformer baselines while significantly reducing inference latency and memory footprint, offering a compelling alternative for scalable language modeling.

Currently, prominent Transformer architectures applied on graphs and meshes for shape analysis tasks employ traditional attention layers that heavily utilize spectral features requiring costly eigenvalue decomposition-based methods. To encode the mesh structure, these methods derive positional embeddings, that heavily rely on eigenvalue decomposition based operations, e.g. on the Laplacian matrix, or on heat-kernel signatures, which are then concatenated to the input features. This paper proposes a novel approach inspired by the explicit construction of the Hodge Laplacian operator in Discrete Exterior Calculus as a product of discrete Hodge operators and exterior derivatives, i.e. (L := star_0^{-1} d_0^T star_1 d_0). We adjust the Transformer architecture in a novel deep learning layer that utilizes the multi-head attention mechanism to approximate Hodge matrices star_0, star_1 and star_2 and learn families of discrete operators L that act on mesh vertices, edges and faces. Our approach results in a computationally-efficient architecture that achieves comparable performance in mesh segmentation and classification tasks, through a direct learning framework, while eliminating the need for costly eigenvalue decomposition operations or complex preprocessing operations.

Polarization analysis in x-ray spectroscopy provides an orientation dependent sensitivity to local bonding environments. For a cluster of atoms, polarization sensitivity is most often discussed through the lens of point group symmetries. However, this is a discrete, qualitative method of classifying clusters, and it does little to indicate the degree of spectral anisotropy. Here we adopt a random forest model for quantitative prediction of spectral anisotropy. Its input relies on simplified local geometric and chemical information that can be obtained from any crystal structure file. The model is trained on over 10,000 experimentally realized transition metal structures from the Materials Project, with the target being VtC-XES calculated using the real space Green's function code FEFF. We find that the model can strongly predict the degree of spectral anisotropy, with the primary factors being derived from the spatial moments of ligands in a cluster.

The push to train ever larger neural networks has motivated the study of initialization and training at large network width. A key challenge is to scale training so that a network's internal representations evolve nontrivially at all widths, a process known as feature learning. Here, we show that feature learning is achieved by scaling the spectral norm of weight matrices and their updates like texttt{fan-out/fan-in}, in contrast to widely used but heuristic scalings based on Frobenius norm and entry size. Our spectral scaling analysis also leads to an elementary derivation of maximal update parametrization. All in all, we aim to provide the reader with a solid conceptual understanding of feature learning in neural networks.

Loss explosions in training deep neural networks can nullify multi-million dollar training runs. Conventional monitoring metrics like weight and gradient norms are often lagging and ambiguous predictors, as their values vary dramatically across different models and even between layers of the same model, making it difficult to establish a unified standard for detecting impending failure. We introduce Spectral Alignment (SA), a novel, theoretically-grounded metric that monitors the distributional alignment between layer inputs and the principal singular vectors of weight matrices. We show that a collapse in the sign diversity of this alignment is a powerful early predictor of representational collapse and training divergence. Empirical results on language models demonstrate that monitoring the SA distribution provides a significantly earlier and clearer warning of loss explosions than traditional scalar metrics. SA's low computational overhead makes it a practical tool for safeguarding model training.

Spectral bias is an important observation of neural network training, stating that the network will learn a low frequency representation of the target function before converging to higher frequency components. This property is interesting due to its link to good generalization in over-parameterized networks. However, in low dimensional settings, a severe spectral bias occurs that obstructs convergence to high frequency components entirely. In order to overcome this limitation, one can encode the inputs using a high frequency sinusoidal encoding. Previous works attempted to explain this phenomenon using Neural Tangent Kernel (NTK) and Fourier analysis. However, NTK does not capture real network dynamics, and Fourier analysis only offers a global perspective on the network properties that induce this bias. In this paper, we provide a novel approach towards understanding spectral bias by directly studying ReLU MLP training dynamics. Specifically, we focus on the connection between the computations of ReLU networks (activation regions), and the speed of gradient descent convergence. We study these dynamics in relation to the spatial information of the signal to understand how they influence spectral bias. We then use this formulation to study the severity of spectral bias in low dimensional settings, and how positional encoding overcomes this.