Sparsification of Rectangular Matrices

Abstract

Consider the problem of sparsifying a rectangular matrix with more columns than rows. This means forming linear combinations of the rows, while preserving the rank, such that the result contains as many zero entries as possible. A combinatorial search method is presented which sparsifies a matrix with exponentially many arithmetic operations in the worst case. Moreover, a method is presented which substantially reduces the combinatorial search space if the matrix gives rise to a non-trivial block structure.

Computing Sparse Multiples of Polynomials

“…Techniques similar to those used in [24] prove that the problem of finding sparse vectors in integer lattices is NP-complete, a problem that was stated to be open in [6]. We prove this result in Theorem 4.3.…”

“…One of our original motivations was to understand the complexity of sparse polynomial implicitization: given a function generating zeros, find a sparse polynomial with those zeros (see [8]). The linear algebra formulation in Chapter 2 relates to to the finding the minimum distance of a binary linear code [2,24] as well as finding "sparsifications" of linear systems [6].…”

Computing Sparse Multiples of Polynomials

“…We also conjecture the intractability of some of these problems, based on similar problems in coding theory. Finally, we show that the construction of Vardy (1997) can be used to show the problem of finding the sparsest vector in an integer lattice is NPcomplete, which was conjectured by Egner and Minkwitz (1998).…”

“…The same problem over integers translates into finding the sparsest vector in an integer lattice. It was posed as an open problem in Egner and Minkwitz (1998). Techniques similar to Vardy (1997) prove that this problem is also NP-complete over the integers, a fact proved in Theorem 2.5.…”

“…Sparse multiples can facilitate efficient arithmetic in extension fields (Brent and Zimmermann, 2003) and in designing interleavers for error-correcting codes (Sadjadpour et al, 2001). The linear algebra formulation in Section 2 relates to finding the minimum distance of a binary linear code (Berlekamp et al, 1978;Vardy, 1997) as well as finding "sparsifications" of linear systems (Egner and Minkwitz, 1998).…”