Bondar, G. (AM) – Geometric and Graph Structured Optimal Transport for Learning and Control

We motivate, formulate, and solve several structured optimal transportation (OT) problems for learning and control. The field of OT had its inception in 1781, when Gaspard Monge sought the optimal map to transport mass from a given initial distribution to a desired terminal distribution. Both OT and its entropy-regularized variant, the Schrödinger bridge problem (SBP), have catalyzed significant developments across mathematics. More recently, theoretical and algorithmic developments in OT and SBP have found myriad applications in stochastic optimal control and generative AI. The connecting theme across the problems studied in this dissertation is the presence of structure which can be exploited in their solution — for our work, this structure arises from geometry underlying a solution space or from a graph capturing a correlation structure between distributional data. Our original contributions begin with automated verification and discovery of OT regularity, followed by stochastic learning and control for resource allocation and scheduling in multi-core real-time systems, where the problem has a natural graph structure. We also formulate and solve the problem of finding an optimal graph structure in a distribution-valued dataset where none is present. While graph-structured OT is a natural fit for data-driven stochastic learning problems, the knowledge of system dynamics can impose geometric structure to the problem. In this dissertation, we establish the equivalence between the stochastic optimal control and path space formulations of SBP over a compact connected Lie group. Additionally, we develop, analyze, and validate a novel numerical solution scheme for the SBP with control-affine dynamics. Finally, we investigate the application of Gromov-Wasserstein (GW) OT — a variant of the problem allowing for transport between incomparable spaces — for spatial proteomics.

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