Approximate Real Symmetric Tensor Rank

TL;DR

This paper explores how small perturbations affect the symmetric tensor rank of real tensors, providing theoretical bounds, algorithms, and geometric insights to understand the minimal rank achievable within an epsilon neighborhood.

Contribution

It introduces new theorems and algorithms that offer constructive upper bounds on symmetric tensor rank under perturbations, combining probabilistic and convex geometric methods.

Findings

Provided bounds for tensor rank under perturbations

Developed algorithms for rank approximation

Connected geometric ideas with tensor decomposition

Abstract

We investigate the effect of an $\varepsilon$-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric $d$-tensor $f$, a norm $||.||$ on the space of symmetric $d$-tensors, and $\varepsilon >0$ are given. What is the smallest symmetric tensor rank in the $\varepsilon$-neighborhood of $f$? In other words, what is the symmetric tensor rank of $f$ after a clever $\varepsilon$-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind; we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.