Tail Bounds for Functions of Weighted Tensor Sums Derived from Random Walks on Riemannian Manifolds

TL;DR

This paper develops new tensor inequalities and bounds for weighted tensor sums from random walks on Riemannian manifolds, advancing mathematical tools for analyzing complex tensor and manifold interactions.

Contribution

It introduces novel tensor inequalities using the Mond-Pecaric method and establishes bounds for transition matrices and tensor sums based on spectral information.

Findings

New tensor inequalities derived using the Mond-Pecaric method

Bounds for transition matrix column sums based on spectral data

Tail bounds for weighted tensor sums from random walks on manifolds

Abstract

This paper presents significant advancements in tensor analysis and the study of random walks on manifolds. It introduces new tensor inequalities derived using the Mond-Pecaric method, which enriches the existing mathematical tools for tensor analysis. This method, developed by mathematicians Mond and Pecaric, is a powerful technique for establishing inequalities in linear operators and matrices, using functional analysis and operator theory principles. The paper also proposes novel lower and upper bounds for estimating column sums of transition matrices based on their spectral information, which is critical for understanding random walk behavior. Additionally, it derives bounds for the right tail of weighted tensor sums derived from random walks on manifolds, utilizing the spectrum of the Laplace-Beltrami operator over the underlying manifolds and new tensor inequalities to enhance the understanding of these complex mathematical structures.