傅立叶矩阵和循环矩阵不是刚性的
Fourier and Circulant Matrices are Not Rigid
摘要：
矩阵刚度的概念最早由Valiant于1977年提出。粗略地说，如果一个矩阵的秩不能通过改变少量的元素而显著降低，则该矩阵是刚性的。有广泛的兴趣在刚性矩阵的Valiant表明，在他的MFCS'77文件，刚性可以用来证明算术电路下界。在一个令人惊讶的结果中，Alman和Williams（FOCS'19）表明，被证明是刚性的（实值）Hadamard矩阵实际上不是非常刚性。Dvir和Edelman（\emph{Theory of Computing}，2019）将这一工作扩展到与Hadamard矩阵相关的矩阵族，但在有限域上。在我们的工作中，我们采取了另一个步骤，在这个方向上，并表明，任何阿贝尔群$G$和功能$f:G \rightarrow \mathbb{C}$，矩阵由$M_{xy} = f(x - y)$为$x,y \in G$是不刚性的。特别是，我们得到，复值傅立叶矩阵，循环矩阵，和Toeplitz矩阵都不是刚性的，不能用来执行Valiant的方法来证明电路下界。当我们考虑固定有限域上的矩阵而不是复数时，我们的结果也成立。这补充了Goldreich和Tal最近的结果（\emph{Comp. Complexity}，2018），他们表明Toeplitz矩阵具有非平凡的刚性（但对于Valiant的方法来说还不够）。我们的工作不同于以前的非刚性的结果，这些作品认为矩阵的基础组的对称性的形式$\mathbb{F}_p^n$与$p$固定和$n$趋于无穷大，而在家庭的矩阵，我们研究的基础组的对称性可以是任何阿贝尔群，特别是循环群$\mathbb{Z}_N$，它有非常不同的结构。我们的研究结果还表明，自然的新候选人的刚性矩阵的形式，其对称群是高度非阿贝尔。
英文摘要：
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 extensive interest in rigid matrices as Valiant showed in his MFCS'77 paper that rigidity can be used to prove arithmetic circuit lower bounds. In a surprising result, Alman and Williams (FOCS'19) showed that the (real valued) Hadamard matrix, which was conjectured to be rigid, is actually not very rigid. This line of work was extended by Dvir and Edelman (\emph{Theory of Computing}, 2019) to a family of matrices related to the Hadamard matrix, but over finite fields. In our work, we take another step in this direction and show that for any abelian group $G$ and function $f:G \rightarrow \mathbb{C}$, the matrix given by $M_{xy} = f(x - y)$ for $x,y \in G$ is not rigid. In particular, we get that complex valued Fourier matrices, circulant matrices, and Toeplitz matrices are all not rigid and cannot be used to carry out Valiant's approach to proving circuit lower bounds. Our results also hold when we consider matrices over a fixed finite field instead of the complex numbers. This complements a recent result of Goldreich and Tal (\emph{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 works considered matrices whose underlying group of symmetries was of the form $\mathbb{F}_p^n$ 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 $\mathbb{Z}_N$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
