Finding Dense Clusters via "Low Rank + Sparse" Decomposition

UBk+1V Block Sparse Householder Decomposition

This paper describes Householder reduction of a rectangular sparse matrix to small band upper triangular form. Using block Householder transformations gives good orthogonality, is computationally efficient, and has good potential for parallelization. The algorithm is similar to the standard dense Householder reduction used as part of the usual dense SVD computation. For the sparse algorithm, th...

Lazy Householder Decomposition of Sparse Matrices

This paper describes Householder reduction of a rectangular sparse matrix to small band upper triangular form Bk+1. Bk+1 is upper triangular with nonzero entries only on the diagonal and on the nearest k superdiagonals. The algorithm is similar to the Householder reduction used as part of the standard dense SVD computation. For the sparse “lazy” algorithm, matrix updates are deferred until a ro...

Sparse and Low-Rank Tensor Decomposition

Motivated by the problem of robust factorization of a low-rank tensor, we study the question of sparse and low-rank tensor decomposition. We present an efficient computational algorithm that modifies Leurgans’ algoirthm for tensor factorization. Our method relies on a reduction of the problem to sparse and low-rank matrix decomposition via the notion of tensor contraction. We use well-understoo...

Complex data analytics via sparse, low-rank matrix approximation

Finding Low-rank Solutions of Sparse Linear Matrix Inequalities using Convex Optimization

This paper is concerned with the problem of finding a low-rank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph. Furthermore, we propose three graph-theoretic convex programs to obtain a low...