On the Stochastic Rank of Metric Functions

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.

💡 Research Summary

The paper investigates the rank properties of integral operators whose kernels are metric (distance) functions defined on a manifold, and it does so from a stochastic perspective. The central problem is formulated as follows: given a random sample of (n) points ({x_i}{i=1}^n) drawn independently from a manifold (M), consider the symmetric matrix (D) with entries (D{ij}=d(x_i,x_j)), where (d) is the chosen metric. The question is what is the probability that (D) has full rank (i.e., rank (n)). This probability is denoted (\mathbb{P}_n(d)).

The authors first prove that when the metric (d) is real‑analytic on (M\times M), (\mathbb{P}_n(d)=1) for any finite (n). The proof relies on the fact that a real‑analytic function admits a multivariate Taylor expansion around any point, and the coefficients of this expansion are polynomial in the coordinates. By selecting points in general position, the rows (or columns) of (D) become linear combinations of distinct monomials, which yields a Vandermonde‑type structure. The non‑vanishing of the corresponding determinant follows from classical results on the zero‑measure of analytic sets, guaranteeing that almost every sample produces a nonsingular distance matrix.

Conversely, if the metric is not analytic—e.g., piecewise linear, Lipschitz but not smooth, or a low‑degree polynomial—the probability of full rank can be strictly less than one. The paper introduces a “regularity condition” that captures the essential analytic features needed for full rank, and it quantifies the measure of the “singular set” where the determinant vanishes. In particular, for metrics that are merely continuous or have isolated singularities, the authors show that (\mathbb{P}_n(d)) may decay to zero as (n) grows, reflecting the possibility that the distance matrix collapses onto a lower‑dimensional subspace.

The second major contribution is an application to the inverse problem of recovering a probability distribution (\mu) on a manifold from its covariance field. The covariance field is defined by
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