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 Schrödinger 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 multiple time-points. The solution sought, as in SBP, represents a probability law on the space of paths that is 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.
CITATION STYLE
Chen, Y., Conforti, G., Georgiou, T. T., & Ripani, L. (2019). Multi-marginal Schrödinger Bridges. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 11712 LNCS, pp. 725–732). Springer. https://doi.org/10.1007/978-3-030-26980-7_75
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