What is the probability that a random symmetric tensor is close to rank-one?

TL;DR

This paper calculates the probability that a symmetric tensor is near rank-one by applying differential geometry and random matrix theory, providing explicit formulas for the reach and curvature of the Veronese variety.

Contribution

It introduces a novel geometric approach to estimate the probability of symmetric tensors being close to rank-one, with explicit formulas derived for the Veronese variety.

Findings

Explicit formulas for reach and curvature coefficients of the Veronese variety.

Complete solution to the probability estimation problem.

Exponential decay formula for rational normal curves.

Abstract

We address the general problem of estimating the probability that a real symmetric tensor is close to rank-one tensors. Using Weyl's tube formula, we turn this question into a differential geometric one involving the study of metric invariants of the real Veronese variety. More precisely, we give an explicit formula for its reach and curvature coefficients with respect to the Bombieri-Weyl metric. These results are obtained using techniques from Random Matrix theory and an explicit description of the second fundamental form of the Veronese variety in terms of GOE matrices. Our findings give a complete solution to the original problem. In the case of rational normal curves it leads to a simple formula describing explicitly exponential decay with respect to the degree of the tensor.