A Normal Form Algorithm for Tensor Rank Decomposition

TL;DR

This paper introduces a novel numerical algorithm for tensor rank decomposition that reformulates the problem as a polynomial system, leveraging algebraic geometry tools to improve efficiency and accuracy.

Contribution

The paper develops a new algorithm based on polynomial systems and algebraic geometry, providing effective bounds and conjecturing a general formula for tensor rank decomposition complexity.

Findings

Algorithm outperforms existing methods in accuracy and speed

Provides bounds for tensor formats and ranks

Conjectures a polynomial time complexity for the problem

Abstract

We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We characterize the complexity of our algorithm in terms of an algebraic property of this polynomial system -- the multigraded regularity. We prove effective bounds for many tensor formats and ranks, which are of independent interest for overconstrained polynomial system solving. Moreover, we conjecture a general formula for the multigraded regularity, yielding a (parameterized) polynomial time complexity for the tensor rank decomposition problem in the considered setting. Our numerical experiments show that our algorithm can outperform state-of-the-art numerical algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption.