Upcoming Event: PhD Dissertation Defense

Three problems in data science

Bobby Shi,

Abstract

Much of the success of modern data science algorithms is attributable to strong theoretical guarantees, largely operating under classical assumptions.  However, as the data ecosystem evolves and applications become more complex, it is necessary to adapt by revisiting the fundamentals of the underlying algorithmic primitives.  In modern data science, data is often multimodal and arrives in theoretically nonstandard ways, e.g., with dependent structure or from non-Euclidean geometries. This thesis advances theory for three problems in data science and high-dimensional statistics, addressing modern algorithmic challenges.

In one line of work, we develop new theory of probabilistic constructs under nonstandard assumptions. One project develops concentration inequalities for the largest eigenvalue of an ergodic sum of random matrices with Markov-dependent structure. A second project proves rapid mixing of the proximal sampler based on the log-Laplace transform, adapting to non-Euclidean geometries, by constructing a novel non-Euclidean stochastic localization process.  In another line of work, we study efficient tensor decomposition via moment matrix extension, motivated by questions about the computational complexity of tensor decomposition as the tensor's dimension and rank grow at a specified rate.