论文标题
在深度神经网络中产生的随机矩阵上。高斯案
On Random Matrices Arising in Deep Neural Networks. Gaussian Case
论文作者
论文摘要
本文处理在深层神经网络分析中产生的随机矩阵产物的奇异值的分布。矩阵类似于样品协方差矩阵的产物类似物，但是，重要的区别在于，在统计和随机矩阵理论的标准设置中，种群协方差矩阵现在是随机的，现在是随机的，是随机数据矩阵的某些功能。通过使用自由概率理论的技术，在最近的工作[21]中已经考虑了这个问题。但是，由于自由概率理论涉及与数据矩阵无关的人口矩阵，因此在这种情况下，其适用性需要额外的理由。在数据矩阵的条目是独立的高斯随机变量的假设下，我们通过使用随机矩阵理论的标准技术的版本来介绍这种理由。在随后的论文[18]中，我们将结果扩展到数据矩阵的条目只是具有几个有限矩的独立分布的随机变量。特别是，这扩展了所谓的宏观普遍性在被考虑的随机矩阵上的特性。
The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important difference is that the population covariance matrices, which are assumed to be non-random in the standard setting of statistics and random matrix theory, are now random, moreover, are certain functions of random data matrices. The problem has been considered in recent work [21] by using the techniques of free probability theory. Since, however, free probability theory deals with population matrices which are independent of the data matrices, its applicability in this case requires an additional justification. We present this justification by using a version of the standard techniques of random matrix theory under the assumption that the entries of data matrices are independent Gaussian random variables. In the subsequent paper [18] we extend our results to the case where the entries of data matrices are just independent identically distributed random variables with several finite moments. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.
