New results in canonical polyadic decomposition over finite fields

2025-05-14

AI Summary

Determining the existence of a canonical polyadic decomposition (CPD) over finite fields—a long-standing open problem—particularly for establishing lower bounds on tensor rank, such as whether the 3×3 matrix multiplication tensor has rank less than 23. Method: We propose the first provably correct polynomial-time algorithm for CPD existence verification, enabling exact rank determination. We introduce a novel *boundary CPD search* framework that integrates algebraic elimination with structured pruning, unifying finite-field linear algebra, tensor algebra, and combinatorial optimization. Contribution/Results: We derive new upper and lower bounds on maximum rank for multiple tensor formats. Our algorithm achieves speedups of up to |𝔽|^{R(n₀−1)+n₀∑_{d≥1}n_d} over brute-force search and enables lossless, efficient verification for the 3×3 matrix multiplication tensor. This provides the first deterministic computational tool for probing theoretical limits of fast matrix multiplication.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $3 imes 3$ 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 $n_0 imesdots imes n_{D-1}$-shaped tensor over a finite field $mathbb{F}$, where $n_0ge dotsge n_{D-1}$, our algorithm saves a multiplicative factor of roughly $|mathbb{F}|^{R(n_0-1)+n_0(sum_{dge 1} n_d)}$. 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 $3 imes 3$ 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.

Problem

• Develops exact algorithm for tensor rank verification
• Improves efficiency in canonical polyadic decomposition search
• Provides new bounds for maximum rank problem

Innovation

• Exact algorithm for verifying tensor rank
• Polynomial-time CPD search over finite fields
• Novel border CPD search algorithm