New results in canonical polyadic decomposition over finite fields

TL;DR

This work studies exact CPD and border CPD over finite fields, addressing when a given tensor has a rank-$R$ CPD and how to certify it efficiently. It introduces an exact CPD search (CPD DFS) with pruning and a border-CPD search based on the border ring $\mathbb{F}[x]/(x^H)$, achieving exponential-time, polynomial-space algorithms with provable correctness. The authors derive new upper and lower bounds on the maximum possible tensor rank, including $R(n,n,n)\le \tfrac{23}{32}n^2+O(n)$ and $R(m,n,mn-k)\ge mn-3\sqrt{3k}+O(1)$, and they compute exact maximal ranks for certain small shapes over $\mathbb{F}_2$. Although the rank of the classic $\langle3,3,3\rangle$ MM tensor remains unresolved, the methods and pruners substantially enhance exact CPD search and yield sharper rank-bounds, with implications for the complexity of fast matrix multiplication and tensor-analytic approaches in discrete settings.

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.

Theorems & Definitions (67)

• Theorem 1
• Theorem 2
• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3: "Rref-pruning"
• proof
• Theorem 3: laskowski