Revised: December 18, 2020

Published: December 31, 2020

**Keywords:**matrix rigidity, circulant matrix

**ACM Classification:**F.2.2, F.1.3

**AMS Classification:**68Q17, 68Q15

**Abstract:**
[Plain Text Version]

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).