For a matrix $M$ and a positive integer $r$, the rank $r$ rigidity of $M$ is the smallest number of entries of $M$ which one must change to make its rank at most $r$. There are many known applications of rigidity lower bounds to a variety of areas in complexity theory, but fewer known applications of rigidity upper bounds. In this paper, we use rigidity upper bounds to prove new upper bounds in a few different models of computation. Our results include: $\bullet$ For any $d> 1$, and over any field $\mathbb{F}$, the $N \times N$ Walsh-Hadamard transform has a depth-$d$ linear circuit of size $O(d \cdot N^{1 + 0.96/d})$. This circumvents a known lower bound of $\Omega(d \cdot N^{1 + 1/d})$ for circuits with bounded coefficients over $\mathbb{C}$ by Pudl\'ak (2000), by using coefficients of magnitude polynomial in $N$. Our construction also generalizes to linear transformations given by a Kronecker power of any fixed $2 \times 2$ matrix. $\bullet$ The $N \times N$ Walsh-Hadamard transform has a linear circuit of size $\leq (1.81 + o(1)) N \log_2 N$, improving on the bound of $\approx 1.88 N \log_2 N$ which one obtains from the standard fast Walsh-Hadamard transform. $\bullet$ A new rigidity upper bound, showing that the following classes of matrices are not rigid enough to prove circuit lower bounds using Valiant's approach: $-$ for any field $\mathbb{F}$ and any function $f : \{0,1\}^n \to \mathbb{F}$, the matrix $V_f \in \mathbb{F}^{2^n \times 2^n}$ given by, for any $x,y \in \{0,1\}^n$, $V_f[x,y] = f(x \wedge y)$, and $-$ for any field $\mathbb{F}$ and any fixed-size matrices $M_1, \ldots, M_n \in \mathbb{F}^{q \times q}$, the Kronecker product $M_1 \otimes M_2 \otimes \cdots \otimes M_n$. This generalizes recent results on non-rigidity, using a simpler approach which avoids needing the polynomial method.

中文翻译：

---

Kronecker产品，低深度电路和矩阵刚性

对于矩阵$ M $和一个正整数$ r $，$ M $的等级$ r $刚性是$ M $最小条目数，必须更改该条目才能使其等级最高为$ r $。在复杂性理论中，刚性下限在许多领域都有许多已知的应用，但是刚性上限的已知应用却很少。在本文中，我们使用刚度上限来证明一些不同计算模型中的新上限。我们的结果包括：$ \ bullet $对于任何$ d> 1 $，并且在任何字段$ \ mathbb {F} $上，$ N \ times N $ Walsh-Hadamard变换具有一个深度-$ d $线性电路，大小为$ O（d \ cdot N ^ {1 + 0.96 / d}）$。对于使用Pudl \'ak（2000）的有界系数超过$ \ mathbb {C} $的电路，它使用系数来绕过$ \ Omega（d \ cdot N ^ {1 + 1 / d}）$的已知下界的多项式，单位为$ N $。我们的构造还推广到由任何固定的2×2矩阵的Kronecker幂给出的线性变换。$ \ bullet $ $ N \ times $$ Walsh-Hadamard变换具有大小为$ \ leq（1.81 + o（1））N \ log_2 N $的线性电路，在$ \ approx 1.88 N \ log_2的范围内有所改善从标准快速Walsh-Hadamard变换获得的N $。$ \ bullet $一个新的刚度上限，表明以下类别的矩阵不够僵硬，无法使用Valiant的方法证明电路的下界：$-$用于任何字段$ \ mathbb {F} $和任何函数$ f：\ {0,1 \} ^ n \ to \ mathbb {F} $，矩阵$ V_f \ in \ mathbb {F} ^ {2 ^ n \ times 2 ^ n} $由任何$ x，y \在\ {0,1 \} ^ n $中，$ V_f [x，y] = f（x \ wedge y）$，以及$-$对于任何字段$ \ mathbb {F} $和任何固定大小的矩阵$ M_1，\ ldots，M_n \ in \ mathbb {F} ^ {q \ times q} $，Kronecker产品$ M_1 \ otimes M_2 \ otimes \ cdots \ otimes M_n $。这使用一种避免需要多项式方法的更简单方法，将有关非刚度的最新结果归纳为一个整体。