Improved upper bounds for the rigidity of Kronecker products

Abstract

The rigidity of a matrix $A$ for target rank $r$ is the minimum number of entries of $A$ that need to be changed in order to obtain a matrix of rank at most $r$. At MFCS'77, Valiant introduced matrix rigidity as a tool to prove circuit lower bounds for linear functions and since then this notion received much attention and found applications in other areas of complexity theory. The problem of constructing an explicit family of matrices that are sufficiently rigid for Valiant's reduction (Valiant-rigid) still remains open. Moreover, since 2017 most of the long-studied candidates have been shown not to be Valiant-rigid. Some of those former candidates for rigidity are Kronecker products of small matrices. In a recent paper (STOC'21), Alman gave a general non-rigidity result for such matrices: he showed that if an $n\times n$ matrix $A$ (over any field) is a Kronecker product of $d\times d$ matrices $M_1,\dots, M_k$ (so $n=d^k$) $(d\ge 2)$ then changing only $n^{1+\varepsilon}$ entries of $A$ one can reduce its rank to $\le n^{1-\gamma}$, where $1/\gamma$ is roughly $2^d/\varepsilon^2$. In this note we improve this result in two directions. First, we do not require the matrices $M_i$ to have equal size. Second, we reduce $1/\gamma$ from exponential in $d$ to roughly $d^{3/2}/\varepsilon^2$ (where $d$ is the maximum size of the matrices $M_i$), and to nearly linear (roughly $d/\varepsilon^2$) for matrices $M_i$ of sizes within a constant factor of each other. As an application of our results we significantly expand the class of Hadamard matrices that are known not to be Valiant-rigid; these now include the Kronecker products of Paley-Hadamard matrices and Hadamard matrices of bounded size.