Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections

TL;DR

This paper proves the existence of efficient block projections for large fields, enabling faster algorithms for sparse matrix inversion, null space basis, rank certification, and determinant computation with probabilistic guarantees.

Contribution

It establishes the correctness of block projections over large fields and derives improved probabilistic algorithms for key matrix computations.

Findings

Expected inversion of sparse matrices in softO(n^{2.27}) operations.

Null space basis and rank certification in softO(n^{2.27}) operations.

Determinant and Smith form algorithms in softO(n^{2.66}) operations.

Abstract

Block projections have been used, in [Eberly et al. 2006], to obtain an efficient algorithm to find solutions for sparse systems of linear equations. A bound of softO(n^(2.5)) machine operations is obtained assuming that the input matrix can be multiplied by a vector with constant-sized entries in softO(n) machine operations. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections, and this has been conjectured. In this paper we establish the correctness of the algorithm from [Eberly et al. 2006] by proving the existence of efficient block projections over sufficiently large fields. We demonstrate the usefulness of these projections by deriving improved bounds for the cost of several matrix problems, considering, in particular, ``sparse'' matrices that can be be multiplied by a vector using softO(n) field operations. We show how to compute the inverse of a sparse matrix over a field F using an expected number of softO(n^(2.27)) operations in F. A basis for the null space of a sparse matrix, and a certification of its rank, are obtained at the same cost. An application to Kaltofen and Villard's Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an integer matrix yields algorithms requiring softO(n^(2.66)) machine operations. The derived algorithms are all probabilistic of the Las Vegas type.