Probabilistic Rank and Matrix Rigidity

We consider a notion of probabilistic rank and probabilistic sign-rank of a matrix, which measures the extent to which a matrix can be probabilistically represented by low-rank matrices. We demonstrate several connections with matrix rigidity, communication complexity, and circuit lower bounds, including: The Walsh-Hadamard Transform is Not Very Rigid. We give surprising upper bounds on the rigidity of a family of matrices whose rigidity has been extensively studied, and was conjectured to be highly rigid. For the 2^n × 2^n Walsh-Hadamard transform H_n (a.k.a. Sylvester matrices, or the communication matrix of Inner Product mod 2), we show how to modify only 2^ϵ n entries in each row and make the rank drop below 2^n(1-Ω(ϵ^2/log(1/ϵ))), for all ϵ > 0, over any field. That is, it is not possible to prove arithmetic circuit lower bounds on Hadamard matrices, via L. Valiant's matrix rigidity approach. We also show non-trivial rigidity upper bounds for H_n with smaller target rank. Matrix Rigidity and Threshold Circuit Lower Bounds. We give new consequences of rigid matrices for Boolean circuit complexity. We show that explicit n × n Boolean matrices which maintain rank at least 2^(log n)^1-δ after n^2/2^(log n)^δ/2 modified entries would yield a function lacking sub-quadratic-size AC^0 circuits with two layers of arbitrary linear threshold gates. We also prove that explicit 0/1 matrices over ℝ which are modestly more rigid than the best known rigidity lower bounds for sign-rank would imply strong lower bounds for the infamously difficult class THR∘ THR.