Complete Decomposition of Symmetric Tensors in Linear Time and Polylogarithmic Precision

Abstract

We study symmetric tensor decompositions, i.e., decompositions of the form $T = \sum _{i=1}^r u_i^{\otimes 3}$ where T is a symmetric tensor of order 3 and $u_i \in \mathbb {C}^n$. In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from the $u_i$. In this paper we assume that the $u_i$ are linearly independent. This implies $r \le n$, i.e., the decomposition of T is undercomplete. We will moreover assume that $r=n$ (we plan to extend this work to the case $r<n$ in a forthcoming paper.) We give a randomized algorithm for the following problem: given T, an accuracy parameter $\epsilon$, and an upper bound B on the condition number of the tensor, output vectors $u'_i$ such that $||u_i - u'_i|| \le \epsilon$ (up to permutation and multiplication by phases) with high probability. The main novel features of our algorithm are:

• We provide the first algorithm for this problem that works in the computation model of finite arithmetic and requires only poly-logarithmic (in n, B and $\frac{1}{\epsilon }$) many bits of precision.

• Moreover, this is also the first algorithm that runs in linear time in the size of the input tensor. It requires $O(n^3)$ arithmetic operations for all accuracy parameters $\epsilon = \frac{1}{\text {poly}(n)}$.

In order to obtain these results, we rely on a mix of techniques from algorithm design and algorithm analysis. The algorithm is a modified version of Jennrich’s algorithm for symmetric tensors. In terms of algorithm design, our main contribution lies in replacing the usual appeal to resolution of a linear system of equations [5, 12] by a matrix trace-based method. The analysis of the algorithm depends on the following components:

• 1. We use the fast and numerically stable diagonalisation algorithm from [1]. We provide better guarantees for the approximate solution returned by the diagonalisation algorithm when the input matrix is diagonalisable.

• 2. We show strong anti-concentration bounds for certain families of polynomials when the randomness is sampled uniformly from a discrete grid.