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 $u_i\in C^n$. 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 ε, and an upper bound B on the condition number of the tensor, output vectors $u_i^{\prime }$ such that $\left|\right|u_i-u_i^{\prime }\left|\right|\le \epsilon$ (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 $O\left(n^3\right)$ arithmetic operations for all accuracy parameters $\epsilon =\frac{1}{\text{poly}\left(n\right)}$.

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 simultaneous diagonalisation 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 [30], [7] 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 [9]. 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.