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Complexity Lower Bounds of Small Matrix Multiplication over Finite Fields via Backtracking and Substitution

Chengu Wang

TL;DR

It is proved that the bilinear complexity of multiplying two $3 \times 3$ matrices over $\mathbb{F}_2$ is at least $20$, improving the longstanding lower bound of $19$ (Blaser 2003).

Abstract

We introduce a new method for proving bilinear complexity lower bounds for matrix multiplication over finite fields. The approach combines the substitution method with a systematic backtracking search over linear restrictions on the first matrix $A$ in the product $AB = C^T$. We enumerate restriction classes up to symmetry; for each class we either obtain a rank lower bound by classical arguments or branch further via the substitution method. The search is organized by dynamic programming on the restricted matrix $A$. As an application we prove that the bilinear complexity of multiplying two $3 \times 3$ matrices over $\mathbb{F}_2$ is at least $20$, improving the longstanding lower bound of $19$ (Bläser 2003). The proof is found automatically within 1.5 hours on a laptop and verified in seconds.

Complexity Lower Bounds of Small Matrix Multiplication over Finite Fields via Backtracking and Substitution

TL;DR

It is proved that the bilinear complexity of multiplying two matrices over is at least , improving the longstanding lower bound of (Blaser 2003).

Abstract

We introduce a new method for proving bilinear complexity lower bounds for matrix multiplication over finite fields. The approach combines the substitution method with a systematic backtracking search over linear restrictions on the first matrix in the product . We enumerate restriction classes up to symmetry; for each class we either obtain a rank lower bound by classical arguments or branch further via the substitution method. The search is organized by dynamic programming on the restricted matrix . As an application we prove that the bilinear complexity of multiplying two matrices over is at least , improving the longstanding lower bound of (Bläser 2003). The proof is found automatically within 1.5 hours on a laptop and verified in seconds.
Paper Structure (21 sections, 4 theorems, 3 equations, 3 algorithms)

This paper contains 21 sections, 4 theorems, 3 equations, 3 algorithms.

Table of Contents

  1. Previous Lower Bounds
  2. Our Results
  3. Preliminaries
  4. Symmetries
  5. Restrictions
  6. A Toy Example: $\mathbf{R}(\langle 2,2,2\rangle)\ge 7$ over $\mathbb{F}_2$
  7. Rank Lower Bound Techniques
  8. Flattening
  9. Forced Product
  10. Degenerate Reduction
  11. Backtracking
  12. Optimization: Multi-threading
  13. Optimization: Restriction Map
  14. Optimization: Local Map and Memory Management
  15. Optimization: Incremental Target Lower Bound
  16. ...and 6 more sections

Key Result

Theorem 1

$\mathbf{R}(\langle 3,3,3\rangle)\ge 20$ over $\mathbb{F}_2$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1: Flattening
  • Lemma 2: Lemma 2 in hopcroft1971minimizing