[Paper Review] Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion

Aymeric Baradat, Hugo Lavenant|arXiv (Cornell University)|Nov 2, 2021

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper establishes a rigorous duality link between regularized unbalanced optimal transport (RUOT) and the branching Schrödinger problem, where the latter is formulated as entropy minimization with respect to a branching Brownian motion (BBM). It proves that the value of RUOT is the lower semicontinuous envelope of the branching Schrödinger problem's value, and provides a correspondence between their solutions, with convergence to partial optimal transport in the small noise limit.

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.

Motivation & Objective

• To establish a theoretical connection between regularized unbalanced optimal transport (RUOT) and the branching Schrödinger problem, a variant of the classical Schrödinger problem with a branching diffusion process.
• To show that the value function of RUOT arises as the lower semicontinuous envelope of the value function of the branching Schrödinger problem.
• To characterize probability laws with finite relative entropy with respect to a branching Brownian motion, enabling a precise correspondence between solutions of the two problems.
• To analyze the small noise limit of the branching Brownian motion, demonstrating convergence to partial optimal transport in the dynamical formulation.
• To provide a numerical framework for solving the dynamical RUOT problem via convex optimization after discretization.

Proposed method

• Formulate the branching Schrödinger problem as entropy minimization with respect to the law of a branching Brownian motion (BBM), under fixed initial and final marginal constraints.
• Use duality arguments to relate the value of the branching Schrödinger problem to the value of the RUOT problem, showing that the latter is the lower semicontinuous envelope of the former.
• Introduce a generalized Itô formula and stochastic calculus for processes with jumps to characterize laws of finite entropy relative to BBM.
• Construct modified BBMs with predictable, space-time-dependent branching and diffusion rates to model the entropy-minimizing dynamics.
• Prove that any law of finite entropy with respect to a BBM must itself be a BBM, establishing a one-to-one correspondence between competitors in the two problems.
• Discretize the dynamical formulation of RUOT and solve it numerically via convex optimization, leveraging the duality and correspondence results.

Experimental results

Research questions

• RQ1 How is the value of regularized unbalanced optimal transport (RUOT) related to the value of the branching Schrödinger problem?
• RQ2 Can laws of finite relative entropy with respect to a branching Brownian motion be characterized, and what structure do they possess?
• RQ3 What happens to the branching Schrödinger problem in the small noise limit, and how does it relate to optimal transport models?
• RQ4 Is there a one-to-one correspondence between solutions of the branching Schrödinger problem and those of the RUOT problem?
• RQ5 How can the dynamical RUOT problem be numerically solved using the theoretical framework developed in the paper?

Key findings

• The value of the regularized unbalanced optimal transport problem is the lower semicontinuous envelope of the value of the branching Schrödinger problem.
• Any probability law with finite relative entropy with respect to a branching Brownian motion must itself be a branching Brownian motion with a specific predictable intensity.
• In the small noise limit, the optimal transport model derived from the branching Schrödinger problem converges to a partial optimal transport problem.
• A one-to-one correspondence exists between solutions of the branching Schrödinger problem and those of the RUOT problem, under appropriate conditions.
• The dynamical formulation of RUOT can be discretized and solved numerically via convex optimization, leveraging the duality and structural results.
• The branching Brownian motion is the natural reference process for unbalanced optimal transport because it yields the closest possible model to optimal transport among measure-valued branching processes.