For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained by the tensor CP decomposition. For the higher rank case, the situation is not as straightforward. In this work we present an algorithm based on a greedy pro...

The paper presents several results that address a fundamental question in low-rank matrices recovery: how many measurements are needed to recover low rank matrices? We begin by investigating the complex matrices case and show that 4nr−4r generic measurements are both necessary and sufficient for the recovery of rank-r matrices in C by algebraic tools developed in [10]. Thus, we confirm a conjec...

1. Abstract. This article presents a fast dense solver for hierarchically off-diagonal low-rank (HODLR) matrices. This solver uses algebraic techniques such as the adaptive cross approximation (ACA) algorithm to construct the low-rank approximation of the off-diagonal matrix blocks. This allows us to apply the solver to any dense matrix that has an off-diagonal low-rank structure without any pr...

While the performance of Robust Principal Component Analysis (RPCA), in terms of the recovered low-rank matrices, is quite satisfactory to many applications, the time efficiency is not, especially for scalable data. We propose to solve this problem using a novel fast incremental RPCA (FRPCA) approach. The low rank matrices of the incrementally-observed data are estimated using a convex optimiza...

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized n × r matrices as well as certain quantized factor algebras M q (n) of Mq(n) are analyzed. For r = 1, . . . , n − 1, M q (n) is the quantized function algebra of rank r matrices obtained by working modulo the ideal generated by all (r+1)×(r+1) quantum subdeterminants and a certai...

In this paper we describe how one can represent a unitary rank structured matrix in an efficient way as a product of unitary or Givens transformations. We provide also some basic operations for manipulating the representation, such as the transition to zerocreating form, the transition to a unitary/Givens-weight representation, as well as an internal pull-through process of the two branches of ...

A well known property of anM-matrixA is that the inverse is element-wise non-negative, which we write asA−1 0. In this paper we consider perturbations ofM-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. The bounds are written in terms of decay estimates which characterize the decay (along rows) of the elements of the inverse matrix. We obtain results f...

Given an n × n matrix, its principal rank characteristic sequence is a sequence of length n + 1 of 0s and 1s where, for k = 0, 1, . . . , n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such att...

7 Given an n × n matrix, its principal rank characteristic sequence is a sequence of 8 length n + 1 of 0s and 1s where, for k = 0, 1, . . . , n, a 1 in the kth position indicates 9 the existence of a principal submatrix of rank k and a 0 indicates the absence of such 10 a submatrix. The principal rank characteristic sequences for symmetric matrices over 11 various fields are investigated, with ...

Matrices with very few nonzero entries cannot have large rank. On the other hand matrices without any zero entries can have rank as low as 1. These simple observations lead us to our main question. For matrices over finite fields, what is the relationship between the rank of a matrix and the number of nonzero entries in the matrix? This question motivated a summer research project collaboration...