Quadratic Lower Bounds on Matrix Rigidity

Abstract

The rigidity of a matrix A with respect to the rank bound r is the minimum number of entries of A that must be changed to reduce the rank of A to or below r. It is a major unsolved problem (Valiant, 1977) to construct “explicit” families of n × n matrices of rigidity n ^1 + δ for r=εn, where ε and δ are positive constants. In fact, no superlinear lower bounds are known for explicit families of matrices for rank bound r=Ω(n).

In this paper we give the first optimal, Ω(n ²), lower bound on the rigidity of two “somewhat explicit” families of matrices with respect to the rank bound r=cn, where c is an absolute positive constant. The entries of these matrix families are (i) square roots of n ² distinct primes and (ii) primitive roots of unity of prime orders for the first n ² primes. Our proofs use an algebraic dimension concept introduced by Shoup and Smolensky (1997) and a generalization of that concept.