Approximate Low-Rank Decomposition for Real Symmetric Tensors

We investigate the effect of an $\varepsilon$-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 $\varepsilon >0$ error tolerance with respect to $||.||$ 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 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 $L_p$-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 $\varepsilon$-room of tolerance. We also report on our experiments with a preliminary implementation of the energy increment algorithm.