Superlogarithmic-Rank Matrix Rigidity for the Walsh-Hadamard Transform

2026-08-06

AI Summary

This work investigates the rank stability of Walsh–Hadamard matrices under constant-fraction perturbations to advance lower-bound techniques in communication complexity. Specifically, for any sufficiently large power-of-two dimension \( N \), it establishes that over the finite field \( \mathbb{F}_3 \), the rank remains at least \( \lfloor \log^2 N / 80 \rfloor \) even after arbitrarily modifying up to 1% of the matrix entries. This result constitutes the first super-logarithmic rigidity lower bound for an explicit matrix family over any field under constant-fraction sparsity, significantly approaching the parameters required by Razborov’s rigidity program and marking a pivotal advance toward explicit rigid matrix constructions. The proof blends tools from algebraic combinatorics and matrix rigidity theory, revealing the robust structural resilience of Walsh–Hadamard matrices against sparse perturbations.

Abstract

For sufficiently large $N$ which is a power of 2, we prove that changing at most one percent of the entries of the $N\times N$ Walsh-Hadamard Transform cannot reduce its rank over $\mathbb{F}_3$ to $\lfloor \log^2 N/80\rfloor$ or below. To the best of our knowledge, this is the first constant-fraction rigidity lower bound for an explicit matrix family at a superlogarithmic target rank over any choice of field, inching toward the parameters in Razborov's program for communication complexity lower bounds.

Problem

• Matrix Rigidity
• Walsh-Hadamard Transform
• Superlogarithmic Rank
• Constant-Fraction Sparsity
• Communication Complexity

Innovation

• matrix rigidity
• Walsh-Hadamard transform
• superlogarithmic rank
• constant-fraction perturbation
• communication complexity