Description
This book marks a milestone in the theory and algorithms of the Schrödinger bridge and its generalizations a topic of significant current interest across statistical mechanics, stochastic control theory, and generative AI. The topic had its genesis in physics, specifically in 1931 when Erwin Schrödinger conceived a thought experiment to interpret quantum mechanics through the lens of non-equilibrium Markov processes. Schrödinger s idea was much ahead of its time: neither the axiomatic theory of stochastic processes, nor the mathematics of large deviations principle were available then. The connections with stochastic control became clear only in the late twentieth century. In the current time, the development of diffusion models in generative AI have re-energized this area via vigorous interdisciplinary research. Simply put, Schrödinger bridge is a diffusion process that interpolates probability distributions observed at given times with maximum likelihood guarantee. This book contributes to two significant generalizations of the Schrödinger bridge: one where prior knowledge in underlying physics is incorporated as some prescribed drift and diffusion coefficients, the other where additional regularization is enforced on the paths of the diffusion process. For the former, this book establishes contraction guarantees for convergence of numerical algorithms. For the latter, this book shows how regularization manifests as reaction rate and derives corresponding Markov kernels for nontrivial models of practical interest. New points of contact with quantum mechanics, in the spirit of Schrödinger s original work, are also uncovered.
Introduction.- The Contraction Coefficient in Linear SBPs.- Classical SBP with Quadratic State Cost.- Linear SBP with Quadratic State Cost.- Generalized SBP with State Cost.- Future Directions and Summary.
Dr. Alexis M.H. Teter received her Ph.D. in Applied Mathematics from the University of California, Santa Cruz, under the supervision of her thesis advisor Professor Abhishek Halder. She received a B.S. in Applied Mathematics with a specialization in computing from the University of California, Los Angeles. Dr. Teter s research interests include optimization, optimal transport, and stochastic control. While completing her doctoral research, Dr. Teter investigated far-reaching generalizations of the Schrödinger bridge problem, originally formulated by physicist Erwin Schrödinger in 1931 1932 in an attempt to understand quantum mechanics through stochastic calculus. During her final summer as a Ph.D. student, Dr. Teter interned at Lawrence Livermore National Laboratory (LLNL), working within the Physics Division. In addition to her research interests, Dr. Teter has passionate interests in STEM teaching with experience in innovative pedagogy, course design, and instructing mathematics and machine learning to middle and high-schoolers and undergraduates.
