## Center for Applied Mathematics Colloquium

Abstract: The need to reason about uncertainty in large, complex, and multi-modal datasets has become increasingly common. The ability to transform samples from one distribution P to another distribution Q enables the solution to many problems in operations research. Performing such transformations, in general, still comprises computational difficulties, especially in high dimensions. Here, we consider the problem of computing such "measure transport maps" efficiently and at scale. Under the mild assumptions that P need not be known but can be sampled from, that the density of Q is known up to a proportionality constant, and that Q is log-concave, we provide a convex optimization formulation. This approach, involving sequentially solving quadratically regularized relative entropy minimization problems, is inspired by variational formulations in non-equilibrium thermodynamics and allows for sequential construction of transport maps with exponential convergence in relative entropy. We provide an empirical risk minimization formulation using maps parametrized by polynomial chaos along with an iterative and convergent algorithm that has been efficiently implemented in GPUs and Amazon Web Services architectures. We provide examples of this framework, within the context of healthcare applications pertaining, to Bayesian inference, Thompson sampling for online decision making, and generative modeling.

Bio: Todd P. Coleman received B.S. degrees in electrical engineering (summa cum laude), as well as computer engineering (summa cum laude) from the University of Michigan. He received M.S. and Ph.D. degrees from MIT in electrical engineering (minor in mathematics), and did postdoctoral studies at MIT in neuroscience. He is currently a Professor in the Department of Bioengineering at UCSD, where he directs the Neural Interaction Laboratory. Dr. Coleman’s research is very multi-disciplinary, using tools from applied probability, physiology, and bioelectronics. His research spans from developing fundamental information theory and machine learning techniques to partnering with clinicians and solving important healthcare challenges. He has been selected as a National Academy of Engineering Gilbreth Lecturer, as a TEDMED speaker, and as a Fellow of the American Institute for Medical and Biological Engineering.