Improved upper bounds for the rigidity of Kronecker products

Bohdan Kivva

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 $r$. Matrix rigidity was introduced by Valiant in 1977 as a tool to prove circuit lower bounds for linear functions and since then this notion has also found applications in other areas of complexity theory. Recently (arXiv 2021), Alman proved that for any field $\mathbb{F}$, $d\geq 2$ and arbitrary matrices $M_1, \ldots, M_n \in \mathbb{F}^{d\times d}$, one can get a $d^n\times d^n$ matrix of rank $\le d^{(1-\gamma)n}$ over $\mathbb{F}$ by changing only $d^{(1+\varepsilon) n}$ entries of the Kronecker product $M = M_1\otimes M_2\otimes \ldots\otimes M_n$, 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), and to nearly linear (roughly $d/\varepsilon^2$) for matrices $M_i$ of sizes within a constant factor of each other. For the case of matrices of equal size, our bound matches the bound given by Dvir and Liu (\textit{Theory of Computing, 2020}) for the rigidity of generalized Walsh--Hadamard matrices (Kronecker powers of DFT matrices), and improves their bounds for DFT matrices of abelian groups that are direct products of small groups.

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