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…

Abstract: Regression discontinuity design (RDD) is a quasi-experimental impact evaluation method ubiquitous in the social- and applied health sciences. It aims to estimate average treatment effects of policy interventions by exploiting jumps in outcomes induced by cut-off assignment rules. Here, we establish a correspondence between the RDD setting and free discontinuity problems, in particular the celebrated Mumford-Shah model in image segmentation. The Mumford-Shah model is non-convex and hence admits local solutions in general. We circumvent this issue by relying on…