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Matrix Multiplication Reductions

Authors Ashish Gola, Igor Shinkar , Harsimran Singh

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Ashish Gola
  • Simon Fraser University, Burnaby, Canada
Igor Shinkar
  • Simon Fraser University, Burnaby, Canada
Harsimran Singh
  • Simon Fraser University, Burnaby, Canada

Acknowledgements

We are grateful to the anonymous referees for their helpful comments. We also thank Sasha Golovnev and Tom Gur for their valuable feedback.

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Ashish Gola, Igor Shinkar, and Harsimran Singh. Matrix Multiplication Reductions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.34

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 ⋅ 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. - 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. - One-sided error in the low agreement regime. Focusing on average case algorithms with one-sided error, we show that over 𝔽₂ 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 ⋅ B on at least 51% of the entries (i.e., has only a slight advantage over the trivial algorithm), and transforms it into an Õ(T) time worst case algorithm, that outputs the correct answer for all inputs with high probability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Matrix Multiplication
  • Reductions
  • Worst case to average case reductions

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