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. 

Neural networks are highly non-linear functions often parametrized by a staggering number of weights. Miniaturizing these networks and implementing them in hardware is a direction of research that is fueled by a practical need, and at the same time connects to interesting mathematical problems. For example, by quantizing, or replacing the weights of a neural network with quantized (e.g., binary) counterparts, massive savings in cost, computation time, memory, and power consumption can be attained. Of course, one wishes to attain these savings while preserving the action of the function on domains of interest. 
We discuss connections to problems in discrepancy theory, present data-driven and computationally efficient stochastic methods for quantizing the weights of already trained neural networks and we prove that our methods have favorable error guarantees under a variety of assumptions.