Volume 16 (2020) Article 20 pp. 1-48
CCC 2019 Special Issue

Fourier and Circulant Matrices are Not Rigid

by Zeev Dvir and Allen Liu

Received: July 7, 2019
Revised: December 18, 2020
Published: December 31, 2020

$ \newcommand{\C}{{\mathbb C}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\F}{{\mathbb{F}}} $

The concept of matrix rigidity was first introduced by Valiant in 1977. Roughly speaking, a matrix is rigid if its rank cannot be reduced significantly by changing a small number of entries. There has been considerable interest in the explicit construction of rigid matrices as Valiant showed in his MFCS'77 paper that explicit families of rigid matrices can be used to prove lower bounds for arithmetic circuits.

In a surprising recent result, Alman and Williams (FOCS'19) showed that the $2^n\times 2^n$ Walsh--Hadamard matrix, which was conjectured to be rigid, is actually not very rigid. This line of work was extended by Dvir and Edelman (Theory of Computing, 2019) to a family of matrices related to the Walsh--Hadamard matrix, but over finite fields. In the present paper we take another step in this direction and show that for any abelian group $G$ and function $f: G \rightarrow \C$, the $G$-circulant matrix, given by $M_{xy} = f(x - y)$ for $x,y \in G$, is not rigid over $\C$. Our results also hold if we replace $\C$ with a finite field $\F_q$ and require that $\gcd(q,|G|) = 1$. En route to our main result, we show that circulant and Toeplitz matrices (over finite fields or $\C$) and Discrete Fourier Transform (DFT) matrices (over $\C$) are not sufficiently rigid to carry out Valiant's approach to proving circuit lower bounds. This complements a recent result of Goldreich and Tal (Comp. Complexity, 2018) who showed that Toeplitz matrices are nontrivially rigid (but not enough for Valiant's method). Our work differs from previous non-rigidity results in that those papers considered matrices whose underlying group of symmetries was of the form $\Z_p^n$ with $p$ fixed and $n$ tending to infinity, while in the families of matrices we study, the underlying group of symmetries can be any abelian group and, in particular, the cyclic group $\Z_N$, which has very different structure. Our results also suggest natural new candidates for rigidity in the form of matrices whose symmetry groups are highly non-abelian.

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A preliminary version of this paper appeared in the Proceedings of the 34th IEEE Conference on Computational Complexity, 2019 (CCC'19).

DOI: 10.4086/toc.2020.v016a020

© 2020 Zeev Dvir and Allen Liu
Licensed under a Creative Commons Attribution License (CC-BY)