Approximate Real Symmetric Tensor Rank

Abstract

We investigate the effect of an ε-room of perturbation tolerance on symmetric tensor decomposition from an algorithmic perspective. More precisely, we prove theorems and design algorithms for the following problem: Suppose a real symmetric d-tensor f , a norm • on the space of symmetric d-tensors, and ε > 0 error tolerance with respect to • are given. What is the smallest symmetric tensor rank in the ε-neighborhood of f ? In other words, what is the symmetric tensor rank of f after a clever ε-perturbation? We provide two different theoretical bounds and three algorithms for approximate symmetric tensor rank estimation. Our first result is a randomized energy increment algorithm for the case of Lp-norms. Our second result is a simple sampling-based algorithm, inspired by some techniques in geometric functional analysis, that works for any norm. We also provide a supplementary algorithm in the case of the Hilbert-Schmidt norm. All our algorithms come with rigorous complexity estimates, which in turn yield our two main theorems on symmetric tensor rank with ε-room of tolerance. We also report on our experiments with a preliminary implementation of the energy increment algorithm.