Low Rank Matrix Rigidity: Tight Lower Bounds and Hardness Amplification

2025-02-26

AI Summary

This work addresses the long-standing open problem of low-rank rigidity for explicit matrices—including Walsh–Hadamard and distance matrices—defined as the minimum number of entries that must be altered to reduce an $N imes N$ matrix to rank at most $r$. Rigidity bears fundamental implications for circuit complexity and communication lower bounds. The authors establish the first tight rigidity lower bound in the regime $r < log N$: they prove that the $N imes N$ Walsh–Hadamard matrix satisfies $R(c_1 log N) geq N^2 (1/2 - N^{-c_2})$, yielding the best-known explicit rigidity bound to date. They introduce a novel hardness amplification framework that lifts moderate-rank rigidity lower bounds into Razborov-style complexity breakthroughs. Technically, the approach integrates combinatorial matrix analysis, structural characterizations of Kronecker and Majority powers, and algebraic perturbation arguments over finite fields.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For an $N imes N$ 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 $N imes N$ Walsh-Hadamard matrix $H_n$ satisfies $$mathcal{R}_{H_n}(c_1 log N) geq N^2 left( frac12 - N^{-c_2} ight),$$ 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 $$mathcal{R}_{H_n}(log^{1 + varepsilon} N) geq N^2 left( frac12 - N^{-1/2^{(log log N)^{o(1)}}} ight)$$ 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.

Problem

• Determine matrix rigidity for explicit families
• Prove new low-rank rigidity lower bounds
• Amplify hardness for higher rank rigidity

Innovation

• Low rank matrix rigidity bounds
• Hardness amplification techniques
• Walsh-Hadamard transform analysis