# Lower Bounds for Matrix Factorization

## TL;DR

This paper constructs explicit families of matrices that cannot be factored into a small number of sparse matrices, establishing stronger lower bounds for matrix factorization and linear circuit complexity.

## Contribution

It provides the first subexponential-time deterministic construction of matrices with high sparsity lower bounds for fixed depth circuits, improving previous super-linear bounds.

## Key findings

- Constructed matrices with lower bounds of n^{1+1/(2d)} for depth-d circuits.
- Improved lower bounds over previous super-linear results.
- Outlined a derandomization approach for stronger bounds.

## Abstract

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 subexponential time of a family $\{M_n\}$ of $n \times n$ matrices which cannot be expressed as a product $M_n = A_1 \cdots A_d$ where the total sparsity of $A_1,\ldots,A_d$ is less than $n^{1+1/(2d)}$. In other words, any depth-$d$ linear circuit computing the linear transformation $M_n\cdot x$ has size at least $n^{1+\Omega(1/d)}$. This improves upon the prior best lower bounds for this problem, which are barely super-linear, and were obtained by a long line of research based on the study of super-concentrators (albeit at the cost of a blow up in the time required to construct these matrices). 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.