Efficiently Correcting Matrix Products

TL;DR

This paper introduces randomized and deterministic algorithms for efficiently correcting erroneous entries in the product of two matrices, significantly improving correction time especially when errors are sparse.

Contribution

It presents the first randomized and deterministic algorithms with improved running times for correcting matrix products with errors over a ring.

Findings

Randomized correction algorithm runs in n^2 + kn ilde{O} time.

Deterministic correction algorithm runs in kn^2 ilde{O} time.

Algorithms are effective for sparse error correction in matrix products.

Abstract

We study the problem of efficiently correcting an erroneous product of two $n\times n$ matrices over a ring. Among other things, we provide a randomized algorithm for correcting a matrix product with at most $k$ erroneous entries running in $\tilde{O}(n^2+kn)$ time and a deterministic $\tilde{O}(kn^2)$-time algorithm for this problem (where the notation $\tilde{O}$ suppresses polylogarithmic terms in $n$ and $k$).