New results in canonical polyadic decomposition over finite fields

TL;DR

This paper introduces an exact, polynomial-time algorithm for canonical polyadic decomposition over finite fields, enabling provable verification of tensor rank and advancing understanding of matrix multiplication complexity.

Contribution

The authors develop a novel exact algorithm for CPD over finite fields that significantly reduces search complexity and can verify tensor rank, addressing longstanding open problems.

Findings

Algorithm saves a multiplicative factor of roughly ||^{R(n_0-1)+n_0(_{d eq 0})} in search complexity.

Algorithm runs in polynomial time, enabling provable verification of tensor rank.

New bounds for maximum tensor rank are established for various tensor shapes.

Abstract

Canonical polyadic decomposition (CPD) is at the core of fast matrix multiplication, a computational problem with widespread implications across several seemingly unrelated problems in computer science. Much recent progress in this field has used randomized heuristic search to find new CPDs, often over a finite field. However, if these techniques fail to find a CPD with low enough rank, they cannot prove that no such CPD exists. Consequently, these methods fail to resolve certain long-standing questions, such as whether the tensor corresponding to $3\times 3$ matrix multiplication has rank less than 23. To make progress on these problems, we develop a novel algorithm that preserves exactness, i.e. they can provably verify whether or not a given tensor has a specified rank. Compared to brute force, when searching for a rank-$R$ CPD of a $n_0\times\dots\times n_{D-1}$-shaped tensor over a finite field $\mathbb{F}$, where $n_0\ge \dots\ge n_{D-1}$, our algorithm saves a multiplicative factor of roughly $|\mathbb{F}|^{R(n_0-1)+n_0(\sum_{d\ge 1} n_d)}$. Additionally, our algorithm runs in polynomial time. We also find a novel algorithm to search border CPDs, a variant of CPDs that is also important in fast matrix multiplication. Finally, we study the maximum rank problem and give new upper and lower bounds, both for families of tensor shapes and specific shapes. Although our CPD search algorithms are still too slow to resolve the rank of $3\times 3$ matrix multiplication, we are able to utilize them in this problem by adding extra search pruners that do not affect exactness or increase asymptotic running time.