Fast computation of the rank profile matrix and the generalized Bruhat decomposition

Abstract

The row (resp. column) rank profile of a matrix describes the stair-case shape of its row (resp. column) echelon form. We describe a new matrix invariant, the rank profile matrix, summarizing all information on the row and column rank profiles of all the leading sub-matrices. We show that this normal form exists and is unique over a field but also over any principal ideal domain and finite chain ring. We then explore the conditions for a Gaussian elimination algorithm to compute all or part of this invariant, through the corresponding PLUQ decomposition. This enlarges the set of known elimination variants that compute row or column rank profiles. As a consequence a new Crout base case variant significantly improves the practical efficiency of previously known implementations over a finite field. With matrices of very small rank, we also generalize the techniques of Storjohann and Yang to the computation of the rank profile matrix, achieving an (r ω + mn) 1+o(1) time complexity for an m × n matrix of rank r, where ω is the exponent of matrix multiplication. Finally, we give connections to the Bruhat decomposition, and several of its variants and generalizations. Consequently, the algorithmic improvements made for the PLUQ factorization, and their implementation, directly apply to these decompositions. In particular, we show how a PLUQ decomposition revealing the rank profile matrix also reveals both a row and a column echelon form of the input matrix or of any of its leading sub-matrices, by a simple post-processing made of row and column permutations.

Time and space efficient generators for quasiseparable matrices

“…It proposes two new structured representations for quasiseparable matrices, a Recursive Rank Revealing (RRR) representation that can be viewed as a simplified version of the HSS representation of Chandrasekaran et al (2006), and a representation based on the generalized Bruhat decomposition, which we name Compact Bruhat (CB) representation. The later one, is made possible by the connection that we make between the notion of quasiseparability and a matrix invariant, the rank profile matrix, that we introduced in Dumas et al (2015) and applied to the generalized Bruhat decomposition in Dumas et al (2016). More precisely, we show that the lower and upper triangular parts of a quasiseparabile matrix have a Generalized Bruhat decompositions off of which many coefficients can be shaved.…”

“…We therefore propose here a compact variation on it, called the compact Bruhat, that will be used to derive algorithms taking advantage of fast matrix multiplication. This structured representation relies on the generalized Bruhat decomposition described in Manthey and Helmke (2007), thanks to the connection with the rank profile matrix made in Dumas et al (2016).…”

“…Theorem 16 (Generalized Bruhat decomposition (Manthey and Helmke, 2007;Dumas et al, 2016)). For any m × n matrix A of rank r, there exist an m × r matrix C in column echelon form, an r × n matrix E in row echelon form, and an r × r permutation matrix R such that A = CRE.…”

Elimination-based certificates for triangular equivalence and rank profiles

“…If the Prover is honest, I is the row rank profile of A and J is the column rank profile of A. Then, the application of Protocol 10 will output the correct rank profile matrix of A I,J which will lead the Verifier to the correct rank profile matrix of A, as described in [8,Theorem 37]. Note that one only needs to verify the lower bound on the rank of A once, which is why Certificate 9 is fully executed once, while the second run only verifies that the committed rank profile is a rank profile indeed.…”

“…The rank profile matrix of A, denoted by R A is the unique m × n {0, 1}-matrix with r nonzero entries, of which every leading sub-matrix has the same rank as the corresponding sub-matrix of A. It is possible to compute R A with a deterministic algorithm in O(mnr ω−2 ) or with a Monte-Carlo probabilistic algorithm in (r ω + m + n + µ(A)) 1+o (1) field operations [8], where µ(A) is the worst case arithmetic cost to multiply A by a vector.…”