We derive estimates for the largest and smallest singular values of sparse rectangular N×n random matrices, assuming limN,n→∞n N=y∈(0,1). consider a model with sparsity parameter pN such that NpN∼logαN some α>1, assume moments matrix elements satisfy condition E|Xjk|4+δ≤C<∞. also entries matrices we are truncated at level (NpN)1 2−ϰ ϰ:=δ 2(4+δ).

The completion of matrices with missing values under the rank constraint is a non-convex optimization problem. A popular convex relaxation is based on minimization of the nuclear norm (sum of singular values) of the matrix. For this relaxation, an important question is whether the two optimization problems lead to the same solution. This question was addressed in the literature mostly in the ca...

An approximate Spielman-Teng theorem for the least singular value sn(Mn) of a random n × square matrix Mn is statement following form: there exist constants C, c > 0 such that all ? 0, Pr(sn(Mn) ? ?) ? nC? + exp(?nc). The goal this paper to develop simple and novel framework proving results discrete matrices. As an application, we prove {0, 1}-valued matrices, each whose rows independent vector...