🤖 AI Summary
This work addresses the complete (i.e., rank-$n$) decomposition of third-order symmetric tensors: given a tensor $T = sum_{i=1}^n u_i^{otimes 3}$ formed from linearly independent vectors ${u_i}_{i=1}^n subset mathbb{C}^n$, the goal is to recover ${u_i}$ up to permutation and unit-modulus phase rotations, with $ell_2$-accuracy $varepsilon$, with high probability. We propose the first randomized algorithm for finite-precision arithmetic that achieves numerically stable, high-probability recovery under the assumption that the tensor’s condition number is bounded above by $B$. The algorithm requires only $O(n^3)$ arithmetic operations and $mathrm{polylog}(n, B, 1/varepsilon)$ bits of precision. Its core innovations integrate tensor spectral analysis, condition-number-aware stability design, and explicit handling of phase invariance in complex vector recovery—thereby breaking the prior trade-off between numerical accuracy and computational complexity.
📝 Abstract
We study symmetric tensor decompositions, i.e. decompositions of the input symmetric tensor T of order 3 as sum of r 3rd-order tensor powers of u_i where u_i are vectors 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 that r is at most 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 where r is strictly less than 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 and u'_i differ by at most epsilon (in the l_2 norm and up to permutation and multiplication by phases) with high probability. The main novel features of our algorithm are: (1) 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 1/epsilon) many bits of precision. (2) 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 = 1/poly(n).
