CAREER: Random Matrices, Schrodinger Bridges, Optimal Transport, and Generative Modeling

This project aims to develop a unifying theory for random matrices with prescribed margins, Schrödinger bridges, and optimal transport, with direct applications in the development of novel generative AI approaches for multimodal data. The framework addresses the challenge that many large systems afford knowledge from distinct perspectives but lack the full picture. Traffic planners, for instance, may know how many cars leave each neighborhood each morning and how many arrive in each part of downtown, but not which routes connect where. They see the totals along the edges of a table, while the entries inside, which tie the two views together, remain hidden. The same challenge exists at the frontier of artificial intelligence: a system may be asked to turn sketches into photographs, may have seen many sketches and many photographs, but no examples pairing the two, and must discover for itself how the two correspond. Faced with such a hidden picture, one can fill it in two natural ways: at random, in the most unbiased way consistent with what we see, or by design, so that the route connecting the two views is as efficient as possible. A central message of this project is that these two answers — one shaped by chance, the other by optimality — turn out to be remarkably close, with deeply intertwined structure. Building on this insight, the project develops a unified mathematical framework connecting random and optimal transport, leading to new methods for generative modeling. The project will also support graduate and undergraduate research training, outreach to high schools and community colleges, and cross-disciplinary workshops. This project develops a unified theory connecting two largely separate bodies of work: random matrices with prescribed row and column sums (modeling contingency tables, transportation problems, and random graphs) and the linked framework of static Schrödinger bridges and entropic optimal transport. Its central insight is that such margin-constrained random matrices are governed by a hidden variational principle, the static Schrödinger bridge. The project pursues three directions: (1) developing the foundational theory of a generalized Schrödinger bridge, including its Kantorovich duality and the convergence of associated Sinkhorn-type algorithms; (2) establishing probabilistic techniques that transfer results from tractable, exponentially tilted models to otherwise intractable margin-constrained random matrix ensembles; and (3) turning these advances into new generative methods, including multi-modal image generation via Schrödinger bridges with domain-adaptive divergences and random transport. The project is in turn expected to yield more flexible, theoretically grounded algorithms for generative AI. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.