Title: Learning and inference for distributions: from optimal transport to MCMC

Date: Friday, July 5th, 2024

Time: 9:30 am - 12:00 pm EST (9:30 pm - 12:00 am Hong Kong Time)

Location: Zoom meeting https://gatech.zoom.us/j/91660072023

Jiaojiao Fan

Machine Learning PhD Student

School of Aerospace Engineering
Georgia Institute of Technology

Committee

1 Prof. Yongxin Chen (School of Aerospace Engineering, Georgia Tech; Advisor)

2 Prof. Molei Tao (School of Mathematics, Georgia Tech)

3 Prof. Peng Chen (School of Computational Science and Engineering, Georgia Tech)

4 Prof. Haomin Zhou (School of Mathematics, Georgia Tech)

5 Prof. Jiaming Liang (Goergen Institute for Data Science & Department of Computer Science, University of Rochester)

Abstract

In machine learning, distributional data are ubiquitous and pivotal across diverse fields. This dissertation tackles large-scale challenges associated with distributional data, which are often defined by either the volume of data or their high dimensionality, particularly in learning and inference contexts.

We explore two fundamental mathematical tools essential for the transformation and manipulation of distributional data: optimal transport (OT) and Markov Chain Monte Carlo (MCMC) sampling. OT, a centuries-old mathematical framework, is used for comparing probability distributions but is often hindered by high computational costs. We enhance the computational efficiency of OT by focusing on two aspects: improving multi-marginal OT with graph structures, and scaling OT to handle millions of samples through the introduction of neural OT solvers.

MCMC sampling, while being the primary method for drawing samples from a given probability density, faces scalability issues, especially in higher dimensions. To overcome these limitations, we introduce a novel algorithm based on the proximal sampler, a type of Gibbs sampling method, and rigorously demonstrate its superior computational efficiency in converging to the target distribution.