Tensor Programs: A Swiss-Army Knife for Nonlinear Random Matrix Theory of Deep Learning and Beyond

Abstract: The resurgence of neural networks has revolutionized artificial intelligence since 2010. Luckily for mathematicians and statistical physicists, the study of large random network scaling limits, which can be thought of as *nonlinear* random matrix theory, is both practically important and mathematically interesting. We describe several problems in this setting and develop a new comprehensive framework, called “tensor
programs,” for solving these problems. This framework can be thought of as an automatic tool to derive the behavior of computation graphs with large matrices, as used in neural network computation. It is very general, and from it we also obtain new proofs of the semicircle and the Marchenko-Pastur laws. Thus, “tensor programs” is broadly useful to linear
and nonlinear random matrix theory alike, and we hope it will be adopted as a standard tool.
This talk presents the work arXiv:1902.04760.