In double asymptotic limits, the squared singular value process of non-square matrix products obeys geometric Dyson Brownian motion whose T-transform solves a Burgers equation, producing the free log-normal law via free multiplicative convolution.
Universality in Deep Neural Networks: An approach via the Lindeberg exchange principle
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abstract
We consider the infinite-width limit of a fully connected deep neural network with general weights, and we prove quantitative general bounds on the $2$-Wasserstein distance between the network and its infinite-width Gaussian limit, under appropriate regularity assumptions on the activation function. Our main tool is a Lindeberg principle for Deep Neural Networks, which we use to successively replace the weights on each layer by Gaussian random variables.
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Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Product of Random Matrices
In double asymptotic limits, the squared singular value process of non-square matrix products obeys geometric Dyson Brownian motion whose T-transform solves a Burgers equation, producing the free log-normal law via free multiplicative convolution.