Abstract

Matrix rigidity is a notion put forth by Valiant as a means for proving arithmetic circuit lower bounds. A matrix is rigid if it is far, in Hamming distance, from any low rank matrix. Despite decades of efforts, no explicit matrix rigid enough to carry out Valiant's plan has been found. Recently, Alman and Williams showed, contrary to common belief, that the $2^n \times 2^n$ Hadamard matrix could not be used for Valiant's program as it is not sufficiently rigid. In this note we observe a similar `non rigidity' phenomena for any $q^n \times q^n$ matrix $M$ of the form $M(x,y) = f(x+y)$, where $f:F_q^n \to F_q$ is any function and $F_q$ is a fixed finite field of $q$ elements ($n$ goes to infinity). The theorem follows almost immediately from a recent lemma of Croot, Lev and Pach which is also the main ingredient in the recent solution of the cap-set problem.

Highlights

  • We begin by defining the notion of matrix rigidity—a property of matrices that combines combinatorial conditions (Hamming distance) with algebraic ones

  • Recall that the Hamming distance between two vectors x, y ∈ Σn over some alphabet Σ is equal to the number of entries i ∈ [n] for which xi = yi

  • RFM(r) is equal to the smallest number of entries in M that one needs to change in order to reduce the rank of M to r

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Summary

Introduction

We begin by defining the notion of matrix rigidity—a property of matrices that combines combinatorial conditions (Hamming distance) with algebraic ones (matrix rank). Our main result is a similar non-rigidity theorem for any qn × qn matrix M of the form M(x, y) = f (x + y), where f : Fnq → Fq is any function and Fq is a fixed finite field of q elements (n goes to infinity).

Results
Conclusion

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