PhD Thesis Defense

Many driving factors of physical systems are often latent or unobserved. Thus, understanding such systems crucially relies on accounting for the influence of the latent structure. This talk describes advances in three aspects of latent-variable modeling: inference, algorithms, and applications. Concretely, motivated by obtaining accurate and interpretable statistical model of the California reservoir system, we focus on two key challenges that arise :

Methods to address these challenges would be useful for reservoir modeling and more broadly across the data sciences. With respect to the first challenge, we describe a geometric reformulation of the notion of a discovery, which enables the development of model selection methodology for a broader class of problems. We highlight the utility of this viewpoint in problems involving latent-variable modeling and low-rank estimation, with a specific algorithm to control for false discoveries in these settings. With respect to the second challenge, we develop a framework, based on Generalized Linear Models, that addresses all these shortcomings. A particularly novel aspect of our formulation is that it incorporates regularizers that are tailored to the type of latent variables -- e.g. max-2 norm for Bernoulli variables, and complete positive norm for Poisson variables -- with a corresponding semidefinite relaxation in each case.