Statistical Estimation under Nonlinear Algebraic Constraints

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Machine learning and artificial intelligence are all around us: new algorithms drive cars, detect melanoma, perform automated surgeries, and are used for scientific discovery. Now, more than ever, there is an urgent need for rigorous understanding and performance guarantees so that we can rely on these new technologies. Many inference questions in machine learning are nonlinear, and the long tradition of linear models in statistics may no longer be applicable. For example, latent variable models correspond to various notions of low-rank structure in tensors; more generally, latent variable graphical models correspond to more complicated tensor decompositions in the discrete case, and other highly nonlinear structure in the continuous case. Additionally, non-parametric models, such as strongly positively dependent random variables, are defined via nonlinear polynomial inequalities. Inference for such models often suffers from high computational and sample size complexity. Designing methods for inference and obtaining convergence guarantees requires a thorough understanding of the underlying nonlinear structure. My research program is on statistical inference, model selection, and testing for models given by nonlinear algebraic constraints. Each of my long-term projects involves problems that are grounded in different fields: statistics, optimization, nonlinear algebra, and combinatorics. This research program consists of three main projects. (1) Total positivity is a powerful notion important in both statistics (where it implies a strong form of positive dependence of random variables) and optimization (where it facilitates dual certificate construction in problems like super-resolution imaging, for example). Total positivity is defined via polynomial inequalities and raises a multitude of open problems across diverse fields, from statistical inference to discrete geometry. (2) Structured tensor decompositions lie at the core of many statistical and scientific problems (such as discrete latent variable models and blind source separation), however, they pose many computational and theoretical challenges. I intend to design algorithms for fast structured decompositions, to study geometric properties of spaces of tensors with certain structure, and to explore new types of tensor decompositions. (3) Causal inference in the presence of hidden variables captures real-world interaction of observed quantities, however it poses a variety of open problems. Here I propose to work on model description and equivalence, as well as model selection problems. In both the discrete and Gaussian cases, the models are semialgebraic sets. Solving the above problems will call for algebraic and combinatorial techniques. To execute this interdisciplinary research program I plan to work with students and young researchers with various skills and interests. Our work together will give each of them a strong foundation for a career in academia or industry.