Any-Dimensional Data Science: Learning, Optimization, and Sampling

Author: Levin, Eitan

Year: 2026

Degree: Dissertation (Ph.D.)

Advisors: Chandrasekaran, Venkat; Tropp, Joel A.

Committee Members: Schulman, Leonard J.; Chandrasekaran, Venkat; Tropp, Joel A.; Wierman, Adam C.

Option: Applied And Computational Mathematics

Abstract

An exciting direction in computational math driven by recent trends in AI involves automating the solution process of entire problem classes. For example, algorithms for sophisticated data analysis tasks are increasingly being learned from training data instead of being designed by hand. There is also increasing interest in computationally producing proofs of mathematical theorems. However, computationally-derived algorithms and proofs, in sharp contrast to manually-designed ones, rarely generalize to inputs of size not seen during training. How can we train a model on inputs of a few small sizes, and generalize it to inputs of any other size? How can we computationally search over proofs of theorems that hold for objects of all sizes? This thesis develops foundations for tackling such any-dimensional problems by using random sampling maps to compare and summarize objects of different sizes. Our framework allows us to systematically derive finitely-parametrized families of functions that take inputs of any size, to prove inequalities between such functions, and to learn them from data. Our methodology leverages new de Finetti-type theorems and the recently-identified phenomenon of representation stability. We illustrate the resulting framework for any-dimensional problems in several applications.