eng
Schloss Dagstuhl ā Leibniz-Zentrum fĆ¼r Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-16
34:1
34:15
10.4230/LIPIcs.APPROX/RANDOM.2024.34
article
Matrix Multiplication Reductions
Gola, Ashish
1
Shinkar, Igor
1
https://orcid.org/0000-0001-5013-6422
Singh, Harsimran
1
https://orcid.org/0009-0006-8131-0263
Simon Fraser University, Burnaby, Canada
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 ā
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 š½ā 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 ā
B on at least 51% of the entries (i.e., has only a slight advantage over the trivial algorithm), and transforms it into an OĢ(T) time worst case algorithm, that outputs the correct answer for all inputs with high probability.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol317-approx-random2024/LIPIcs.APPROX-RANDOM.2024.34/LIPIcs.APPROX-RANDOM.2024.34.pdf
Matrix Multiplication
Reductions
Worst case to average case reductions