Inference via low-dimensional couplings

• Main Authors: Spantini, Alessio (Author), Bigoni, Daniele (Author), Marzouk, Youssef M (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Aeronautics and Astronautics (Contributor)
• Format: Article
• Language: English
• Published: 2020-08-03T13:59:58Z.
• Subjects: Article
• Online Access: Get fulltext

 We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a target measure of interest. Direct simulation from the desired measure can then be achieved by pushing forward reference samples through the map. Yet characterizing such a map-e.g., representing and evaluating it-grows challenging in high dimensions. The central contribution of this paper is to establish a link between the Markov properties of the target measure and the existence of low-dimensional couplings, induced by transport maps that are sparse and/or decomposable. Our analysis not only facilitates the construction of transformations in high-dimensional settings, but also suggests new inference methodologies for continuous non-Gaussian graphical models. For instance, in the context of nonlinear state-space models, we describe new variational algorithms for filtering, smoothing, and sequential parameter inference. These algorithms can be understood as the natural generalization-to the non-Gaussian case-of the square-root Rauch-Tung-Striebel Gaussian smoother.