Geometric Approach For Majorizing Measures on Hadamard Manifolds

TL;DR

This paper introduces a geometric method to establish the majorizing measure theorem for Gaussian processes on Hadamard manifolds by analyzing volume ratios between the index space and its convex hull.

Contribution

It provides a novel geometric approach to prove the majorizing measure theorem specifically for Hadamard manifolds, expanding the theoretical understanding of Gaussian processes.

Findings

Derived an upper bound for volume ratios on Hadamard manifolds

Applied volume ratios to prove the majorizing measure theorem geometrically

Extended the geometric framework to non-Euclidean index spaces

Abstract

Gaussian processes can be treated as subsets of a standard Hilbert space, however, the volume size relation between the underlying index space of random processes and its convex hull is not clear. The understanding of such volume size relations can help us to establish a majorizing measure theorem geometrically. In this paper, we assume that the underlying index space of random processes is a simply connected manifold with sectional curvature less than negative one (Hadamard manifold). We derive the upper bound for the ratio between the volume of the underlying index space and the volume of its convex hull. We then apply this volume ratio to prove the majorizing measure theorem geometrically.