arXiv research

Study Finsler metric measure manifolds' concentration properties.

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 …

The paper provides concentration bounds for kernel matrix eigenvalues.

Bounds on Steklov eigenvalues for manifolds with boundary.

We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point with…

The paper studies concentration of measure on manifolds with boundary, focusing on $1$-Lipschitz functions.

How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove …

Large deviations for fat tailed distributions, i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated on a single variable. The regime of large deviations is separated from the regi…

The paper explores how approximation theory can improve understanding of smooth kernels in machine learning.

The paper finds the largest eigenvalue for a specific type of domain in hyperbolic space.

The paper analyzes the performance of delay-based reservoir computing using eigenvalue analysis.

The paper examines how the first Steklov-Dirichlet eigenvalue changes with the distance between two concentric circles.

We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take th…

We consider principal component analysis (PCA) in decomposable Gaussian graphical models. We exploit the prior information in these models in order to distribute its computation. For this purpose, we reformulate the problem in the sparse inverse covariance (concentration) domain and solve the global eigenvalue problem …

We find a deterministic equivalent for random feature regression's test error, independent of feature map dimension.

The paper proves concentration inequalities for diffusion processes.

Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.

Study LSTD on LQR, finds sample complexity for value function estimation.

The paper explores how kernel eigenalignments affect generalization in KRR.

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…

Let $B_1$ be a ball of radius $r_1$ in $S^n(\Hy^n)$, and let $B_0$ be a smaller ball of radius $r_0$ such that $\bar{B_0}\subset B_1$. For $S^n$ we consider $r_1< π$. Let $u$ be a solution of the problem $-\La u =1$ in $\Om := B_1\setminus \bar{B_0}$ vanishing on the boundary. It is shown that the associated functional…

Detects non-causal artifacts in multivariate regression models.

New insights into spectral statistics of sample covariance matrix for stable linear systems.

From concentration inequalities for the suprema of Gaussian or Rademacher processes an inequality is derived. It is applied to sharpen existing and to derive novel bounds on the empirical Rademacher complexities of unit balls in various norms appearing in the context of structured sparsity and multitask dictionary lear…

Eigenvalue problems for Laplacian in specific domains are studied, showing maximum eigenvalues occur under certain conditions.

The paper solves a specific type of Ambrosetti-Prodi problem with solutions having clustering concentration layers.

Paper identifies key function spaces for ReLU networks based on Fisher information.

Study reveals unique spectral features in neural network optimization.

Analyzes the Hessian of ReLU networks, proving skewed eigenvalue distribution.

New methods improve accuracy in detecting concentric objects.

High-dimensional models become unstable when sample size falls below a critical level, leading to a phase transition.

Develops a game-theoretic approach to solve SGEP efficiently.

Generative model controls heterophily in graph signals.

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 $p \ge \frac{(1/2 + δ) \log n}{n},$ then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around $1$. We est…

Power-law spectrum of random feature model is preserved in neural networks.

Study on stochastic approximation with Polyak-Ruppert averaging for linear systems.

The paper improves count data regression models for overdispersed data.

The paper examines distances and random walks on hyperbolic surfaces.

New method clusters signed graphs using matrix power means.

Unified signSGD and gradient descent analysis for neural networks.

This article concerns upper bounds for $L^\infty$-norms of random approximate eigenfunctions of the Laplace operator on a compact aperiodic Riemannian manifold $(M,g).$ We study $f_λ$ chosen uniformly at random from the space of $L^2$-normalized linear combinations of Laplace eigenfunctions with eigenvalues in the inte…

We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $λ$. The condition that ensures this is that their $L^2$ norms ove…

New algorithm learns LQR with $O(\sqrt{T})$ regret using Langevin dynamics and excitation.

In the celebrated book entitled Metric Structures for Riemannian and Non-Riemannian Spaces, so-called Green Book, Gromov presented a problem regarding a metric measure space. Gromov posed the question Bound the expansion coefficient from below in terms of the observable diameter. The overall aim of the current study is…

The paper studies neural networks with wide layers and finds a deformed semicircle law.

Theory for algebraic data on categories via concentration structures.

The paper advances U-statistics in dependent settings, improving spectral estimation and goodness-of-fit tests.

SCOPE estimator improves covariance and precision matrix estimation.