Low Rank Matrix Rigidity: Tight Lower Bounds and Hardness Amplification
Josh Alman, Jingxun Liang
TL;DR
This work analyzes the rigidity of explicit matrices in the low-rank regime, focusing on the Walsh–Hadamard transform and the distance matrix, and introduces a Boolean rigidity framework that tightly bounds how well a low-rank matrix can approximate these objects. It develops a spectral method to derive nearly tight lower bounds, showing for constant prime $p$ that $\\mathcal{R}_A^{\mathbb{F}_p^{bool}}(r) \ge N^2(\tfrac12 - c^r \sigma_1/N)$, with concrete implications for Kronecker powers and the distance matrix. The paper also proves hardness amplification results demonstrating that modest improvements in low-rank rigidity would imply Razborov rigidity and thus breakthrough lower bounds in communication complexity. Together, these results delineate the power and limits of rigidity-based approaches in constructing small-depth circuits and in proving strong lower bounds for explicit matrices. The findings illuminate why current rigidity-based circuit lower-bounding techniques face intrinsic barriers and guide future directions for either stronger low-rank bounds or alternative methodologies.
Abstract
For an $N \times N$ matrix $A$, its rank-$r$ rigidity, denoted $\mathcal{R}_A(r)$, is the minimum number of entries of $A$ that one must change to make its rank become at most $r$. Determining the rigidity of interesting explicit families of matrices remains a major open problem, and is central to understanding the complexities of these matrices in many different models of computation and communication. We focus in this paper on the Walsh-Hadamard transform and on the `distance matrix', whose rows and columns correspond to binary vectors, and whose entries calculate whether the row and column are close in Hamming distance. Our results also generalize to other Kronecker powers and `Majority powers' of fixed matrices. We prove two new results about such matrices. First, we prove new rigidity lower bounds in the low-rank regime where $r < \log N$. For instance, we prove that over any finite field, there are constants $c_1, c_2 > 0$ such that the $N \times N$ Walsh-Hadamard matrix $H_n$ satisfies $$\mathcal{R}_{H_n}(c_1 \log N) \geq N^2 \left( \frac12 - N^{-c_2} \right),$$ and a similar lower bound for the other aforementioned matrices. This is tight, and is the new best rigidity lower bound for an explicit matrix family at this rank; the previous best was $\mathcal{R}(c_1 \log N) \geq c_3 N^2$ for a small constant $c_3>0$. Second, we give new hardness amplification results, showing that rigidity lower bounds for these matrices for slightly higher rank would imply breakthrough rigidity lower bounds for much higher rank. For instance, if one could prove $$\mathcal{R}_{H_n}(\log^{1 + \varepsilon} N) \geq N^2 \left( \frac12 - N^{-1/2^{(\log \log N)^{o(1)}}} \right)$$ over any finite field for some $\varepsilon>0$, this would imply that $H_n$ is Razborov rigid, giving a breakthrough lower bound in communication complexity.