# Multi-marginal Schrodinger bridges

## TL;DR

This paper extends the classical Schrödinger Bridge Problem to second-order particle dynamics with partial temporal data, formulating it as an optimal control problem with a Fisher information term, and connecting it to measure-valued splines in Wasserstein space.

## Contribution

It introduces a novel second-order Schrödinger bridge formulation with partial observations, incorporating Fisher information and linking to measure-valued splines in Wasserstein space.

## Key findings

- Derived a time-symmetric optimal control formulation with Fisher information.
- Established a connection to measure-valued splines and Sinkhorn-like algorithms.
- Demonstrated potential applications in signal processing and data science.

## Abstract

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.