High-Dimensional Probability for High-Dimensional Data

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Artificial Intelligence is undergoing a revolution that is fueled by empirical successes of deep learning, a class of machine learning methods based on artificial neural networks. These successes reverberate across a broad spectrum of data science problems. Nevertheless, theoretical understanding of deep learning is scarce. This project is aimed at building rigorous mathematical foundations for modern and future approaches to learning from big data. High-dimensional probability is proposed as a natural framework for the mathematical exploration of deep learning. This project has a double benefit. On the one hand, it is aimed at theoretically explaining the successes of deep learning. On the other hand, the project will inspire future theoretical developments in high-dimensional probability, especially in random matrix theory. The project also provides research training opportunities for graduate students. This project will address theoretical problems in high-dimensional probability that are inspired by open problems in data science. A pressing need for mathematical justification is evident in the area of deep learning, whose stunning success on real-world data applications is not theoretically explained yet. This project proposes high-dimensional probability as a natural framework for the mathematical exploration of deep learning. A unifying theme of most of the problems in this proposal is nonlinear random matrix theory, where random matrices are transformed by a nonlinearity, which alters their spectral and geometric behavior. Nonlinearities empower neural networks and quantizers, underlie the concepts of random Boolean threshold functions, random tensors and geometric graphs. Exploring the unusual spectral behavior of pseudolinear and inhomogeoenous random matrices are the main general thrust of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.