Spiky Rank及其在刚性和电路中的应用
Spiky Rank and Its Applications to Rigidity and Circuits
AI总结：
本文提出spiky rank作为新的矩阵复杂度度量，展示了其在矩阵刚性和深度2 ReLU电路中的应用。
摘要：
我们引入了spiky rank，一种新的矩阵参数，通过将后者组合的组合结构与线性代数的灵活性相结合，从而增强了blocky rank。一个spiky矩阵是块结构的，其对角块是任意秩一矩阵，spiky rank是表示一个矩阵为和所需的最小此类矩阵数量。这种度量扩展了实矩阵的blocky rank，并且对于具有组合和代数特征的问题更具鲁棒性。我们的概念贡献如下：我们提出了spiky rank作为一种良好的矩阵复杂度度量，并通过应用展示了其潜力。我们表明，大的spiky rank意味着高的矩阵刚性，并且spiky rank的下界会给出深度2 ReLU电路的下界，这是神经网络的基本构建块。在技术方面，我们为随机矩阵建立了紧界，并开发了显式下界的框架，将其应用于汉明距离矩阵和谱扩展器。最后，我们将spiky rank与其他矩阵参数相关联，包括blocky rank、稀疏性和γ₂-范数。
英文摘要：
We introduce spiky rank, a new matrix parameter that enhances blocky rank by combining the combinatorial structure of the latter with linear-algebraic flexibility. A spiky matrix is block-structured with diagonal blocks that are arbitrary rank-one matrices, and the spiky rank of a matrix is the minimum number of such matrices required to express it as a sum. This measure extends blocky rank to real matrices and is more robust for problems with both combinatorial and algebraic character. Our conceptual contribution is as follows: we propose spiky rank as a well-behaved candidate matrix complexity measure and demonstrate its potential through applications. We show that large spiky rank implies high matrix rigidity, and that spiky rank lower bounds yield lower bounds for depth-2 ReLU circuits, the basic building blocks of neural networks. On the technical side, we establish tight bounds for random matrices and develop a framework for explicit lower bounds, applying it to Hamming distance matrices and spectral expanders. Finally, we relate spiky rank to other matrix parameters, including blocky rank, sparsity, and the $γ_2$-norm.
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。
