2019/02/22 by Yongxin Chen, Chen, Yongxin, Giovanni Conforti +5 · 3 citations
Computer Science · Engineering · Mathematics · #46Nxx #58J65 #93E20 #Anomaly Detection Techniques and Applications #Applied mathematics #Computer science #Dimension (graph theory) #FOS: Mathematics #Flow (mathematics) #Fluid Dynamics and Turbulent Flows #Geometry #Image (mathematics) #Interpolation (computer graphics) #Joint probability distribution #Marginal distribution #Mathematical analysis #Mathematics #Measure (data warehouse) #Optimization and Control (math.OC) #Probability (math.PR) #Probability measure #Pure mathematics #Random variable #Statistical Methods and Inference #math.OC #math.PR #msc:46Nxx #msc:58J65 #msc:93E20
paper · pdf · doi:10.48550/arxiv.1902.08319
published in arXiv (Cornell University) (Cornell University) · 8 pages
arxiv created 2019/02/22 · openalex publication_date 2019/02/22 · arxiv updated 2019/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
We consider the problem to identify the most likely flow in phase space, of (inertial) particles under stochastic forcing, that is in agreement with spatial (marginal) distributions that are specified at a set of points in time. The question raised generalizes the classical Schrodinger Bridge Problem (SBP) which seeks to interpolate two specified end-point marginal distributions of overdamped particles driven by stochastic excitation. While we restrict our analysis to second-order dynamics for the particles, the data represents partial (i.e., only positional) information on the flow at \em multiple time-points. The solution sought, as in SBP, represents a probability law on the space of paths this closest to a uniform prior while consistent with the given marginals. We approach this problem as an optimal control problem to minimize an action integral a la Benamou-Brenier, and derive a time-symmetric formulation that includes a Fisher information term on the velocity field. We underscore the relation of our problem to recent measure-valued splines in Wasserstein space, which is akin to that between SBP and Optimal Mass Transport (OMT). The connection between the two provides a Sinkhorn-like approach to computing measure-valued splines. We envision that interpolation between measures as sought herein will have a wide range of applications in signal/images processing as well as in data science in cases where data have a temporal dimension.