Chernoff Bounds for Tensor Expanders on Riemannian Manifolds Using Graph Laplacian Approximation

TL;DR

This paper develops tensor Chernoff bounds for random walks on Riemannian manifolds by improving graph approximations, enabling probabilistic tail bounds for high-dimensional data on complex geometric structures.

Contribution

It introduces a novel method for constructing graph approximations of manifolds that preserve spectral properties, extending tail bound analysis to high-dimensional manifold-valued data.

Findings

Derived tensor Chernoff bounds for Riemannian manifolds

Ensured spectral similarity between manifolds and graph approximations

Extended tail bounds to high-dimensional, dependent random objects

Abstract

This paper addresses the advancement of probability tail bound analysis, a crucial statistical tool for assessing the probability of large deviations of random variables from their expected values. Traditional tail bounds, such as Markov's, Chebyshev's, and Chernoff bounds, have proven valuable across numerous scientific and engineering fields. However, as data complexity grows, there is a pressing need to extend tail bound estimation from scalar variables to high-dimensional random objects. Existing studies often rely on the assumption of independence among high-dimensional random objects, an assumption that may not always be valid. Building on the work of researchers like Garg et al. and Chang, who employed random walks to model high-dimensional ensembles, this study introduces a more generalized approach by exploring random walks over manifolds. To address the challenges of constructing an appropriate underlying graph for a manifold, we propose a novel method that enhances random walks on graphs approximating the manifold. This approach ensures spectral similarity between the original manifold and the approximated graph, including matching eigenvalues, eigenvectors, and eigenfunctions. Leveraging graph approximation technique proposed by Burago et al. for manifolds, we derive the tensor Chernoff bound and establish its range for random walks on a Riemannian manifold according to the underlying manifold's spectral characteristics.