Abstract: Contemporary generative models used in the context of unsupervised learning have primarily been designed around the construction of a map between two probability distributions that transform samples from the first into samples from the second. Advances in this domain have been governed by the introduction of algorithms or inductive biases that make learning this map, and the Jacobian of the associated change of variables, more tractable. The challenge is to choose what structure to impose on the transport to…

Abstract: Energy-based models are a recent class of probabilistic generative models wherein the distribution being learned is parametrized up to a constant of proportionality (i.e. a partition function). Fitting such models using maximum likelihood (i.e. finding the parameters which maximize the probability of the observed data) is computationally challenging, as evaluating the partition function involves a high dimensional integral. Thus, newer incarnations of this paradigm instead train other losses which obviate the need to evaluate partition functions. Prominent examples include score matching (in which we fit…

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…