Faster Sparse Matrix Inversion and Rank Computation in Finite Fields

TL;DR

This paper presents a faster method for inverting sparse matrices and computing their rank over finite fields, achieving improved expected time complexity by leveraging structured matrix decompositions and low displacement rank techniques.

Contribution

It introduces a novel approach combining block Krylov and Hankel matrix decompositions with low displacement rank methods for efficient sparse matrix inversion and rank computation over finite fields.

Findings

Expected inversion time reduced to O(n^{2.2131})

Improved bounds for structured matrix inversion

Enhanced algorithms in topological data analysis and finite group theory

Abstract

We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the decomposition of an arbitrary matrix into block Krylov and Hankel matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the explicit inverse of a block Hankel matrix using low displacement rank techniques for structured matrices and fast rectangular matrix multiplication algorithms. We generalize our inversion method to block structured matrices with other displacement operators and strengthen the best known upper bounds for explicit inversion of block Toeplitz-like and block Hankel-like matrices, as well as for explicit inversion of block Vandermonde-like matrices with structured blocks. As a further application, we improve the complexity of several algorithms in topological data analysis and in finite group theory.