New methods correct spectral distortions using known analyte concentrations.
problem Distorted spectral shapes from absorbing and scattering contributions.
method Modified penalized baseline correction methods that incorporate known analyte concentrations.
result Improved prediction performance on near infra-red data sets.
We consider the problem of estimating a spectral risk measure (SRM) from i.i.d. samples, and propose a novel method that is based on numerical integration. We show that our SRM estimate concentrates exponentially, when the underlying distribution has bounded support. Further, we also consider the case when the underlyi…
The paper shows how solutions of perturbed Dirac operators concentrate near singular sets.
problem Understanding concentration of solutions for perturbed Dirac operators.
method Analyzing the algebraic criterion on $(c, \A)$ and spectral properties of deformed Laplacians.
result Proves an index localization theorem based on spectral separation properties.
In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative 1-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…
Improved spectral clustering guarantees for dynamic stochastic block models.
problem Analyzing Spectral Clustering in dynamic stochastic block models.
method Extending guarantees to sparse and smooth DSBM, linking sparsity and smoothness.
result Improved error bounds for consistent recovery in dynamic DSBM.
Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing and evaluating linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design for graph signals, lead to …
Study spectral properties of sparse random graphs to recover latent vectors.
problem Recovering latent vectors in sparse random geometric graphs.
method Analyzes spectral concentration and uses orthogonal polynomial expansions, decoupling, and matrix concentration.
result Sharpens spectral norm bounds and proves exact recovery for Gaussian mixture models.
Kernel networks' stability edge linked to Fisher Information singularity.
problem Understanding the stability edge in high-capacity kernel Hopfield networks.
method Statistical manifold analysis and Riemannian geometry.
result The Ridge of Optimization corresponds to the Edge of Stability, revealing a dual equilibrium.
We study the concentration of random kernel matrices around their mean. We derive nonasymptotic exponential concentration inequalities for Lipschitz kernels assuming that the data points are independent draws from a class of multivariate distributions on Rd, including the strongly log-concave distributions u…
Method estimates number of clusters in Block Markov Chain trajectories.
problem Challenges in choosing number of clusters for sequential data.
method Spectral embedding and density-based clustering.
result Asymptotically consistent method for estimating clusters.
Recent work has shown that tight concentration of the entire spectrum of singular values of a deep network's input-output Jacobian around one at initialization can speed up learning by orders of magnitude. Therefore, to guide important design choices, it is important to build a full theoretical understanding of the spe…
Study robust covariance estimation in large data with concentrated vectors.
problem Estimating robust covariance in large data with concentrated vectors.
method Fixed point of a contracting function using stable semi-metric and concentration of measure.
result Existence and uniqueness of robust estimator with evaluated limiting spectral distribution.
Study on signed graphs with random signs, focusing on community detection.
problem Community detection in signed stochastic block models.
method Strong concentration inequalities for adjacency and Laplacian matrices, applied to signed Laplacian matrix.
result The sign of the first eigenvector of the Laplacian matrix defines a weakly consistent estimator for balanced community detection.
SCOPE estimator improves covariance and precision matrix estimation.
problem Estimating covariance and precision matrices accurately.
method Distributionally robust optimization with convex spectral divergence.
result SCOPE estimator reduces spectral bias and improves condition number.
This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermarti…
We identify spectral conditions for reliable neural probe interpretation.
problem Unreliable performance of linear probes in interpreting neural representations.
method Formalized Spectral Identifiability Principle (SIP) based on eigengap and Fisher error.
result Reliability of neural probes depends on the eigengap relative to Fisher estimation error.
Free boundary minimal submanifolds with boundaries on concentric spheres
problem Finding minimal submanifolds with boundaries on concentric spheres in Euclidean space
method Using a Steklov problem with an indefinite weight
result Exact Morse index of an m-dimensional flat annulus in an n-dimensional spherical shell Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for other methods the spectral information can be used to estimate the excess risk. An …
A hybrid model combines diffusion and neural operator methods for stress prediction in hyperelastic materials.
problem Challenges in predicting stress fields in hyperelastic materials with complex microstructures.
method A hybrid surrogate framework combining a conditional denoising diffusion probabilistic model (cDDPM) and a modified DeepONet.
result The hybrid model consistently outperforms traditional methods by one to two orders of magnitude.
The paper extends spectral estimates to hyperbolic surfaces with hyperbolic ends.
problem Proving a necessary condition for observability of the heat semigroup on manifolds.
method Propagation of smallness estimates of Carleman and Logunov-Malinnikova type.
result Established spectral estimates for surfaces with hyperbolic ends, proving the thickness condition is necessary.
We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed δ>0 if p≥n(1/2+δ)logn, then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around 1. We est…
Spectral portfolio theory links neural networks to wealth dynamics via SGD weight matrices.
problem Understanding wealth dynamics from neural network training.
method Direct identification of weight matrices as portfolio allocation matrices, linking SGD forces to portfolio dynamics.
result Spectral properties of SGD weight matrices transition between additive and multiplicative regimes, influencing wealth dynamics.
Unified theory of ownership concentration, overlap, and dependence.
problem Understanding the complex layers of ownership concentration, overlap, and dependence in financial markets.
method Develops a unified quadratic framework for analyzing these layers and their interactions.
result Unified framework shows that the same residual operator measures static overlap and governs linearized market transmission.
Random SNNs are stable and simple, with low-frequency Fourier spectra.
problem Stability and robustness of spiking neural networks.
method Boolean function analysis and Fourier spectrum concentration.
result Random LIF-SNNs are stable and biased towards simple functions.
Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received …
Spectral density matrix estimation of multivariate time series is a classical problem in time series and signal processing. In modern neuroscience, spectral density based metrics are commonly used for analyzing functional connectivity among brain regions. In this paper, we develop a non-asymptotic theory for regularize…
The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curvature Clustering (TSCC) algori…
We consider spectral clustering algorithms for community detection under a general bipartite stochastic block model (SBM). A modern spectral clustering algorithm consists of three steps: (1) regularization of an appropriate adjacency or Laplacian matrix (2) a form of spectral truncation and (3) a k-means type algorithm…
We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspe…
Study spectral properties of sub-Riemannian Laplacians, proving quantum ergodicity and heat kernel asymptotics.
problem Spectral properties of sub-Riemannian Laplacians.
method Quantum ergodicity results, small-time asymptotics of sub-Riemannian heat kernels, Weyl law.
result Weyl law and spectral concentration on Lie brackets of length r-1.
High-dimensional spectroscopy data makes ML models achieve near-perfect accuracy, even when chemical distinctions are absent.
problem Why machine learning models achieve near-perfect accuracy in spectroscopic classification tasks without chemically meaningful features.
method Theoretical analysis grounded in the Feldman-Hajek theorem and concentration of measure, combined with specific experiments on synthetic and real fluorescence spectra.
result Infinitesimal distributional differences in high-dimensional spaces can lead to perfect separability, making models achieve near-perfect accuracy in spectroscopy.
The paper corrects for node degree in spectral clustering using random walk Laplacian.
problem Node degree heterogeneity in spectral clustering.
method Graph spectral embedding using the random walk Laplacian.
result The embedding provides uniformly consistent estimates of degree-corrected latent positions.
New insights into spectral statistics of sample covariance matrix for stable linear systems.
problem Estimating high-dimensional stable state transition matrices from noisy data.
method Combining spectral theorem for non-Hermitian operators, concentration of measure, and perturbation theory.
result The spectral radius of the sample covariance matrix exhibits phase transitions in high dimensions.
New Bethe-Hessian method improves community detection in sparse networks.
problem Detect communities in sparse networks efficiently.
method Spectral clustering using the Bethe-Hessian matrix.
result Bethe-Hessian consistently estimates block number above Kesten-Stigum threshold.
Generative model controls heterophily in graph signals.
problem Controlling heterophily in graph signals for better model effectiveness.
method Combines graphon-based generator with spectral filtering of Gaussian node features.
result Establishes theoretical guarantees for heterophily control and convergence.
We obtain a Bernstein-type inequality for sums of Banach-valued random variables satisfying a weak dependence assumption of general type and under certain smoothness assumptions of the underlying Banach norm. We use this inequality in order to investigate in the asymptotical regime the error upper bounds for the broad …
Paper identifies key function spaces for ReLU networks based on Fisher information.
problem Understanding the structure of Fisher information matrices in ReLU networks.
method Spectral decomposition of Fisher information matrices, focusing on the first three eigenspaces.
result The first three eigenspaces account for 97.7% of the trace of the Fisher information matrix, corresponding to spherical harmonic functions of order ≤2.
Study on identifying and inferring nonlinear dynamics on unknown networks.
problem Identifying network structure in nonlinear dynamic systems with unknown interactions.
method Showed network structure is not generically identified, requiring sufficient spectral heterogeneity. Developed necessary and sufficient conditions for identification and proposed a semiparametric estimator.
result Necessary and sufficient conditions for identification of network structure in nonlinear dynamic systems.
Recently, there has been a surge of interest in using spectral methods for estimating latent variable models. However, it is usually assumed that the distribution of the observations conditioned on the latent variables is either discrete or belongs to a parametric family. In this paper, we study the estimation of an $m…
Study identifies cancer genes through graph anomaly analysis of protein interactions.
problem Insufficient modeling of biological information in protein interaction networks for cancer gene identification.
method Proposes HIerarchical-Perspective Graph Neural Network (HIPGNN) to detect weight heterogeneity and spectral flattening in cancer gene nodes.
result HIPGNN detects weight heterogeneity and spectral flattening, leading to improved cancer gene identification.
In this paper, we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in Rn. Our approach focuses on high dimensions and relies solely on the concentration properties of the components in the mixture. We give several results describing of the structure of kernel matrices …
Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.
problem Proving upper bounds for the width of compact manifolds with boundary.
method Utilizes spacetime harmonic functions, μ-bubbles, and spinorial Callias operators.
result Generalizes Schoen-Yau black hole existence theorem to higher dimensions.
Study improves fractional posterior for 1-bit matrix completion.
problem Estimating a binary matrix from observed entries.
method Fractional posterior approach with low-rank factorization and spectral scaled Student priors.
result Concentration results for fractional posterior, demonstrating effectiveness in matrix recovery.
Spectral algorithm recovers community structure in sparse hypergraphs.
problem Community detection in sparse random hypergraphs with community structure and higher-order interactions.
method Spectral algorithm with three steps: hyperedge selection, spectral partition, and correction/merging.
result Weak consistency achieved for weak signal-to-noise ratio.
The paper studies neural networks with wide layers and finds a deformed semicircle law.
problem Investigating spectral distributions of neural networks in the ultra-wide regime.
method Analyzes empirical kernel matrices, proves deformed semicircle law, provides nonlinear Hanson-Wright inequality.
result Emergence of a deformed semicircle law in the ultra-wide neural network regime.
Graph spectral analysis can yield meaningful embeddings of graphs by providing insight into distributed features not directly accessible in nodal domain. Recent efforts in graph signal processing have proposed new decompositions-e.g., based on wavelets and Slepians-that can be applied to filter signals defined on the g…
In this paper we propose and study a family of continuous wavelets on general domains, and a corresponding stochastic discretization that we call Monte Carlo wavelets. First, using tools from the theory of reproducing kernel Hilbert spaces and associated integral operators, we define a family of continuous wavelets by …
The paper advances U-statistics in dependent settings, improving spectral estimation and goodness-of-fit tests.
problem Non-asymptotic analysis of U-statistics in dependent Markov chain settings.
method Proved new concentration and exponential inequalities for U-statistics, applied to spectral estimation, online algorithms, and goodness-of-fit tests.
result Established new results for spectral estimation, online algorithms, and goodness-of-fit tests in Markov chain settings.