Lower Bounds for Matrix Factorization

Ben Lee Volk*, Mrinal Kumar

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 (n1-Ω(1/d)) of a family { Mn} of n× n matrices which cannot be expressed as a product Mn= A1⋯ Ad where the total sparsity of A1, … , Ad is less than n1+1/(2d). In other words, any depth-d linear circuit computing the linear transformation Mn· x has size at least n1+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.

Original languageEnglish
Article number6
JournalComputational Complexity
Issue number1
StatePublished - Jun 2021
Externally publishedYes


  • 68Q04
  • 68Q15
  • 68Q17
  • Algebraic complexity
  • Linear circuits
  • Lower bounds
  • Matrix factorization


Dive into the research topics of 'Lower Bounds for Matrix Factorization'. Together they form a unique fingerprint.

Cite this