Theory of Computing
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Title : Fourier and Circulant Matrices are Not Rigid
Authors : Zeev Dvir and Allen Liu
Volume : 16
Number : 20
Pages : 1-48
URL : http://www.theoryofcomputing.org/articles/v016a020
Abstract
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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).