XGBoost is one of the most successful machine learning methods in practice, yet its theoretical foundations remain poorly understood. In particular, despite its widespread use, there is currently no rigorous characterization of the function class that XGBoost is capable of learning. In this talk, I will present a theoretical framework that addresses this question. I will introduce an infinite-dimensional function class that extends finite ensembles of bounded-depth regression trees, together with a complexity measure that generalizes the regularization penalty used by XGBoost. I will show that every minimizer of the XGBoost objective is a minimizer of an equivalent penalized regression problem over this larger function class, thereby revealing the function class that XGBoost implicitly targets. I will also discuss a smoothness-based characterization of this function class, connecting XGBoost to classical smoothness-based methods in nonparametric regression. Finally, I will present statistical guarantees showing that least squares estimation over this class achieves nearly minimax-optimal rates of convergence without suffering from the curse of dimensionality. These results provide a theoretical explanation for why XGBoost performs well in practice.

Using an operator-theoretic approach, we provide a unified framework for Optimal Transport (OT) between Gaussian measures on separable Hilbert spaces. This formulation allows us to fully characterize the Monge and Kantorovich problems without imposing any regularity or non-degeneracy conditions on the covariance operators. We then develop the dynamic picture, explicitly characterizing 2-Wasserstein geodesics and particle dynamics in this general setting. Extending these results to Entropic OT, we show that the optimal entropic coupling operates as a precise spectral shrinkage of the correlation operator. Time permitting, I will discuss the algorithmic advantages of this spectral perspective and present complementary viewpoints connecting these transport problems.

Functional principal component analysis (FPCA) is a fundamental tool for exploring variation in samples of random curves or surfaces. We propose a new approach to FPCA for functional data observed irregularly and sparsely over their domains, based on smoothing directly at the level of the eigenfunctions. Our formulation leads to an efficient optimization-based procedure whose computational and storage costs are comparable to those of standard multivariate PCA for regularly observed data. The method is flexible with respect to domain geometry and model class, accommodates structural constraints and penalties, and facilitates uncertainty quantification via resampling and asymptotic theory.