Abstract: The Paley graph is a classical number-theoretic construction of a graph that is believed to behave "pseudorandomly" in many regards. Accurately bounding the clique number of the Paley graph is a long-standing open problem in number theory, with applications to several other questions about the statistics of finite fields. I will present recent results studying the application of convex optimization and spectral graph theory to this problem, which involve understanding the extent to which the Paley graph is "spectrally…

Abstract: Sampling from high-dimensional probability distributions is a fundamental and challenging problem encountered throughout science and engineering. One of the most popular approaches to tackle such problems is the Markov chain Monte Carlo (MCMC) paradigm. While MCMC algorithms are often simple to implement and widely used in practice, analyzing the rate of convergence to stationarity, i.e. the "mixing time", remains a challenging problem in many settings. I will describe a new technique based on pairwise correlations called "spectral independence", which has been…

Abstract: In this talk I will propose new tools for the exploratory data analysis of data objects taking values in a general separable metric space. First, I will introduce depth profiles, where the depth profile of a point ω in the metric space refers to the distribution of the distances between ω and the data objects. I will describe how depth profiles can be harnessed to define transport ranks, which capture the centrality of each element in the metric space with respect to the…

Abstract: Reinforcement learning is the study of models and procedures for optimal sequential decision-making under uncertainty.  At its heart lies the Bellman optimality operator, whose unique fixed point specifies an optimal policy and value function.  In this talk, we discuss two classes of variational methods that can be used to obtain approximate solutions with accompanying error guarantees.  For policy evaluation problems based on on-line data, we present Krylov-Bellman boosting, which combines ideas from Krylov methods with non-parametric boosting.  For policy optimization problems based on…