Error correction in fast matrix multiplication and inverse

TL;DR

This paper introduces new algorithms for detecting and correcting errors in matrix multiplication and inversion over any field, efficiently handling small and large error counts without extra encoding.

Contribution

The paper presents novel error correction algorithms that operate directly on original matrices, improving efficiency and scalability over previous methods.

Findings

Algorithms work efficiently for small errors, scaling well with larger errors.

No additional encoding or information needed beyond original matrices.

Builds on recent advances in matrix product correction and verification techniques.

Abstract

We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs and the erroneous output. Their running time is softly linear in the number of nonzero entries in these matrices when the number of errors is sufficiently small, and they also incorporate fast matrix multiplication so that the cost scales well when the number of errors is large. These algorithms build on the recent result of Gasieniec et al (2017) on correcting matrix products, as well as existing work on verification algorithms, sparse low-rank linear algebra, and sparse polynomial interpolation.