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Matrix Multiplication Reductions

Ashish Gola, Igor Shinkar, Harsimran Singh

TL;DR

The paper tackles worst-case to average-case reductions for matrix multiplication over finite fields by introducing an agreement measure ${\rm agr}$ between an algorithm’s output and the true product $A B$. It delivers two main results: (i) in the high-agreement, two-sided error regime, a self-correction approach converts a predominantly correct average-case algorithm into a worst-case correct solver with running time $O(T(n)\log n)$, and (ii) in the low-agreement, one-sided error regime over ${\rm F}_2$, an additive-combinatorics–driven reduction, leveraging low-rank decompositions and the Bogolyubov-Ruzsa framework, yields a worst-case algorithm with running time $\tilde{O}(T(n))$ that succeeds with high probability. The methods rely on structural tools from additive combinatorics (Chang’s lemma and Bogolyubov-Ruzsa) to identify large affine subspaces where corrections can be performed, enabling robust correction from partial to exact products. The results bridge average-case algorithms and robust worst-case solvers for matrix multiplication, with open questions about extending the high/low-agreement framework to other fields and two-sided settings beyond the presented cases.

Abstract

In this paper we study a worst case to average case reduction for the problem of matrix multiplication over finite fields. Suppose we have an efficient average case algorithm, that given two random matrices $A,B$ outputs a matrix that has a non-trivial correlation with their product $A \cdot B$. Can we transform it into a worst case algorithm, that outputs the correct answer for all inputs without incurring a significant overhead in the running time? We present two results in this direction. (1) Two-sided error in the high agreement regime: We begin with a brief remark about a reduction for high agreement algorithms, i.e., an algorithm which agrees with the correct output on a large (say $>0.9$) fraction of entries, and show that the standard self-correction of linearity allows us to transform such algorithms into algorithms that work in worst case. (2) One-sided error in the low agreement regime: Focusing on average case algorithms with one-sided error, we show that over $\mathbb{F}_2$ there is a reduction that gets an $O(T)$ time average case algorithm that given a random input $A,B$ outputs a matrix that agrees with $A \cdot B$ on at least $51\%$ of the entries (i.e., has only a slight advantage over the trivial algorithm), and transforms it into an $\widetilde{O}(T)$ time worst case algorithm, that outputs the correct answer for all inputs with high probability.

Matrix Multiplication Reductions

TL;DR

The paper tackles worst-case to average-case reductions for matrix multiplication over finite fields by introducing an agreement measure between an algorithm’s output and the true product . It delivers two main results: (i) in the high-agreement, two-sided error regime, a self-correction approach converts a predominantly correct average-case algorithm into a worst-case correct solver with running time , and (ii) in the low-agreement, one-sided error regime over , an additive-combinatorics–driven reduction, leveraging low-rank decompositions and the Bogolyubov-Ruzsa framework, yields a worst-case algorithm with running time that succeeds with high probability. The methods rely on structural tools from additive combinatorics (Chang’s lemma and Bogolyubov-Ruzsa) to identify large affine subspaces where corrections can be performed, enabling robust correction from partial to exact products. The results bridge average-case algorithms and robust worst-case solvers for matrix multiplication, with open questions about extending the high/low-agreement framework to other fields and two-sided settings beyond the presented cases.

Abstract

In this paper we study a worst case to average case reduction for the problem of matrix multiplication over finite fields. Suppose we have an efficient average case algorithm, that given two random matrices outputs a matrix that has a non-trivial correlation with their product . Can we transform it into a worst case algorithm, that outputs the correct answer for all inputs without incurring a significant overhead in the running time? We present two results in this direction. (1) Two-sided error in the high agreement regime: We begin with a brief remark about a reduction for high agreement algorithms, i.e., an algorithm which agrees with the correct output on a large (say ) fraction of entries, and show that the standard self-correction of linearity allows us to transform such algorithms into algorithms that work in worst case. (2) One-sided error in the low agreement regime: Focusing on average case algorithms with one-sided error, we show that over there is a reduction that gets an time average case algorithm that given a random input outputs a matrix that agrees with on at least of the entries (i.e., has only a slight advantage over the trivial algorithm), and transforms it into an time worst case algorithm, that outputs the correct answer for all inputs with high probability.
Paper Structure (16 sections, 8 theorems, 35 equations, 3 algorithms)

This paper contains 16 sections, 8 theorems, 35 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. High agreement regime with two-sided error:
  3. Low agreement with one-sided error:
  4. Related work
  5. Open problems
  6. Low agreement with two-sided error:
  7. Generalizing over finite fields:
  8. Preliminaries
  9. Additive Combinatorics Tools
  10. High Agreement with Two-Sided Error
  11. Correctness:
  12. Running time:
  13. Low Agreement with One-Sided Error
  14. Computing the good coordinates
  15. Proof of \ref{['thm:mm-alg-reduction-one-sided']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Fix a finite field $\mathbb{F}$. Let $\alpha \in (0, 1/8)$. Let ${\mathrm ALG}$ be an algorithm that gets as input two matrices $A,B \in \mathbb{F}^{n \times n}$, runs in time $T(n)$, and outputs a matrix $C \in \mathbb{F}^{n \times n}$. Suppose that Then, there is an algorithm ${\mathrm ALG}^*$ that gets as input two matrices $A,B \in \mathbb{F}^{n \times n}$, runs in time $O(T(n) \cdot \log(n))

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3: Chang's Theorem Chang02
  • Lemma 2.4: Probabilistic quasi-polynomial Bogolyubov-Ruzsa lemma
  • proof
  • Claim 2.5
  • ...and 22 more