AI today can pass the Turing test and is in the process of transforming science, technology, society, humans, and beyond.
Surprisingly, modern AI is built out of two very simple and old ideas, rebranded as deep learning: neural networks and
gradient descent learning. When a typical feed-forward neural network is trained by gradient descent, with an L2 regularizer
to avoid overly large synaptic weights, a strange phenomenon occurs: at the optimum, each neuron becomes "balanced"
in the sense that the L2 norm of its incoming synaptic weights becomes equal to the L2 norm of its outgoing synaptic weights. We develop a theory that explains this phenomenon and exposes its generality. Balance emerges with a variety of activation functions, a variety of regularizers including all Lp regularizers, and a variety of networks including recurrent networks. A simple local balancing algorithm can be applied to any neuron and at any time, instead of just at the optimum. Most remarkably, stochastic iterated application of the local balancing algorithm always converges to a unique, globally balanced, state.

The Erdos-Kac Central Limit Theorem says that if one selects an integer at random from 1 to N, then the number of distinct prime divisors of this number satisfies a Central Limit Theorem. We (the speaker in joint work with Tom Mountford) give new proof of this result using the Riemann zeta distribution and a Tauberian Theorem. The proof generalizes easily to other situations such as polynomials over a finite field or ideals in a number field.

The non-backtracking operator is a non-Hermitian matrix associated with an undirected graph. It has become a powerful tool in the spectral analysis of random graphs. Recently, many new results for sparse Hermitian random matrices have been proved for the corresponding non-backtracking operator and translated into a statement of the original matrices through the Ihara-Bass formula. In another direction, efficient algorithms based on the non-backtracking matrix have successfully reached optimal sample complexity in many sparse low-rank estimation problems. I will talk about my recent work with the help of the non-backtracking operator. This includes the Bai-Yin law for sparse random rectangular matrices, hypergraph community detection, and tensor completion. 

Deep generative models have seen a meteoric rise in capabilities across a wide array of domains, ranging from natural language and vision to scientific applications such as precipitation forecasting and molecular generation. However, a number of important applications focus on data which is inherently infinite-dimensional, such as time-series, solutions to partial differential equations, and audio signals. This relatively under-explored class of problems poses unique theoretical and practical challenges for generative modeling. In this talk, we will explore recent developments for infinite-dimensional generative models, with a focus on diffusion-based methodologies.