非线性函数作用于正交不变矩阵，谱分布趋近高斯等效结果。
Entrywise application of non-linear functions on orthogonally invariant matrices
- 通过逐元素应用非线性函数研究矩阵谱变化
- 渐近下效果等价于线性组合加独立GOE矩阵
- 适用于单/多矩阵情形，含相关性建模
本文研究对对称正交不变随机矩阵集进行逐元素非线性函数变换后，其谱分布的变化。同时考虑多变量情形，即对多个正交不变矩阵的元素施加多变量函数，甚至允许矩阵间的相关性。我们发现，在所有这些情况下，均存在高斯等效原理：非线性函数的渐近效应等同于对所涉矩阵作线性组合，并额外叠加一个独立的高斯正交系综（GOE）矩阵。以单矩阵情形下的ReLU函数和双矩阵情形下的max函数为例，展示了该原理的适用性。
原文摘要 · Abstract (English)
In this article, we investigate how the entrywise application of a non-linear function to symmetric orthogonally invariant random matrix ensembles alters the spectral distribution. We treat also the multivariate case where we apply multivariate functions to entries of several orthogonally invariant matrices; where even correlations between the matrices are allowed. We find that in all those cases a Gaussian equivalence principle holds, that is, the asymptotic effect of the non-linear function is the same as taking a linear combination of the involved matrices and an additional independent GOE. The ReLU-function in the case of one matrix and the max-function in the case of two matrices provide illustrative examples.
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