Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion

Abstract

We consider the problem of minimizing the entropy of a law with respect to the law of a reference branching Brownian motion under density constraints at an initial and final time. We call this problem the branching Schrödinger problem by analogy with the Schrödinger problem, where the reference process is a Brownian motion. Whereas the Schrödinger problem is related to regularized (a.k.a. entropic) optimal transport, we investigate here the link of the branching Schrödinger problem with regularized unbalanced optimal transport. This link is shown at two levels. First, relying on duality arguments, the values of these two problems of calculus of variations are linked, in the sense that the value of the regularized unbalanced optimal transport (seen as a function of the initial and final measure) is the lower semi-continuous relaxation of the value of the branching Schrödinger problem. Second, we also explicit a correspondence between the competitors of these two problems, and to that end we provide a fine description of laws having a finite entropy with respect to a reference branching Brownian motion.We investigate the small noise limit, when the noise intensity of the branching Brownian motion goes to 0: in this case we show, at the level of the optimal transport model, that there is convergence to partial optimal transport. We also provide formal arguments about why looking at the branching Brownian motion, and not at other measure-valued branching Markov processes, like superprocesses, yields the problem closest to optimal transport. Finally, we explain how this problem can be solved numerically: the dynamical formulation of regularized unbalanced optimal transport can be discretized and solved via convex optimization.

Trajectory Inference via Mean-field Langevin in Path Space

“…It turns out that this noise also has a favorable effect on the algorithm we develop here. This model can be extended to allow particles to branch and die [4], this extension is discussed in Section 4.2.…”

“…(1). That is, in time dt the probability of a particle X t dividing is g(t, X t )dt [4]. We would like to incorporate this knowledge, and also allow for additional mass variations to account for the inaccuracy of our prior g.…”

Efficient Preconditioners for Solving Dynamical Optimal Transport via Interior Point Methods

“…Furthermore, it draws a clear link between optimal transport and continuum mechanics, and it is naturally suited for Eulerian discretizations. Finally, it can be easily generalized to other problems by penalizing/constraining the evolution ρ (such as variational mean field games and planning problems [2], or unbalanced optimal transport [16,6]). On the other hand, the numerical solution of (1), or its system of optimality conditions (2), poses significant challenges.…”