Lower Bounds for Matrix Factorization

We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science; the most significant being in the context of lower bounds for algebraic circuits which compute linear transformations, matrix rigidity and data structure lower bounds. We first show, for every constant d, a deterministic construction in time exp (n^1-Ω(1/d)) of a family { M_n} of n× n matrices which cannot be expressed as a product M_n= A₁⋯ A_d where the total sparsity of A₁, … , A_d is less than n^1+1/(2d). In other words, any depth-d linear circuit computing the linear transformation M_n· x has size at least n^1+Ω(1/d). The prior best lower bounds for this problem were barely super-linear, and were obtained by a long line of research based on the study of super-concentrators. We improve these lower bounds at the cost of a blow up in the time required to construct these matrices. Previously, however, such constructions were not known even in time 2 ^O(n) with the aid of an NP oracle. We then outline an approach for proving improved lower bounds through a certain derandomization problem, and use this approach to prove asymptotically optimal quadratic lower bounds for natural special cases, which generalize many of the common matrix decompositions.