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Greedy inference with structure-exploiting lazy maps
Michael Brennan · Daniele Bigoni · Olivier Zahm · Alessio Spantini · Youssef Marzouk

Thu Dec 10 06:30 AM -- 06:45 AM (PST) @ Orals & Spotlights: Probabilistic Models/Statistics

We propose a framework for solving high-dimensional Bayesian inference problems using \emph{structure-exploiting} low-dimensional transport maps or flows. These maps are confined to a low-dimensional subspace (hence, lazy), and the subspace is identified by minimizing an upper bound on the Kullback--Leibler divergence (hence, structured). Our framework provides a principled way of identifying and exploiting low-dimensional structure in an inference problem. It focuses the expressiveness of a transport map along the directions of most significant discrepancy from the posterior, and can be used to build deep compositions of lazy maps, where low-dimensional projections of the parameters are iteratively transformed to match the posterior. We prove weak convergence of the generated sequence of distributions to the posterior, and we demonstrate the benefits of the framework on challenging inference problems in machine learning and differential equations, using inverse autoregressive flows and polynomial maps as examples of the underlying density estimators.

Author Information

Michael Brennan (Massachusetts Institute of Technology)
Daniele Bigoni (Massachusetts Institute of Technology)
Olivier Zahm (INRIA)
Alessio Spantini (Massachusetts Institute of Technology)
Youssef Marzouk (Massachusetts Institute of Technology)

Youssef Marzouk is a Professor in the Department of Aeronautics and Astronautics at MIT, and co-director of the MIT Center for Computational Science & Engineering. He is also a core member of MIT's Statistics and Data Science Center and director of the MIT Aerospace Computational Design Laboratory. His research interests lie at the intersection of computation and statistical inference with physical modeling. He develops new methodologies for uncertainty quantification, Bayesian modeling and computation, data assimilation, experimental design, and machine learning in complex physical systems. His methodological work is motivated by a wide variety of engineering, environmental, and geophysics applications. He received his SB, SM, and PhD degrees from MIT and spent four years at Sandia National Laboratories before joining the MIT faculty in 2009. He is a recipient of the Hertz Foundation Doctoral Thesis Prize, the Sandia Laboratories Truman Fellowship, the US Department of Energy Early Career Research Award, and the Junior Bose Award for Teaching Excellence from the MIT School of Engineering. He is an Associate Fellow of the AIAA and currently serves on the editorial boards of the SIAM Journal on Scientific Computing, the SIAM/ASA Journal on Uncertainty Quantification, and several other journals. He is also an avid coffee drinker and occasional classical pianist.

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