On the Stochastic Rank of Metric Functions

TL;DR

This paper investigates the conditions under which integral operators with metric kernels on manifolds have finite rank, linking stochastic properties of random samples to the rank of distance matrices, with applications to manifold recovery.

Contribution

It provides necessary and sufficient conditions for finite rank of metric kernel operators and addresses the stochastic problem of rank probability in random distance matrices, especially for analytic metrics.

Findings

Analytic metrics ensure full rank probability in random samples.

Conditions for finite rank relate to the properties of the kernel functions.

Applications include manifold recovery from covariance fields.

Abstract

For a class of integral operators with kernels metric functions on manifold we find some necessary and sufficient conditions to have finite rank. The problem we pose has a stochastic nature and boils down to the following alternative question. For a random sample of discrete points, what will be the probability the symmetric matrix of pairwise distances to have full rank? When the metric is an analytic function, the question finds full and satisfactory answer. As an important application, we consider a class of tensor systems of equations formulating the problem of recovering a manifold distribution from its covariance field and solve this problem for representing manifolds such as Euclidean space and unit sphere.