Extremal flows in Wasserstein space

We develop an intrinsic geometric approach to the calculus of variations in theWasserstein space. We show that the flows associated with the Schr\ua8odinger bridge with general prior, with optimal mass transport, and with the Madelung fluid can all be characterized as annihilating the first variation of a suitable action. We then discuss the implications of this unified framework for stochastic mechanics: It entails, in particular, a sort of fluid-dynamic reconciliation between Bohm\u2019s and Nelson\u2019s stochastic mechanics

About the analogy between optimal transport and minimal entropy

We describe some analogy between optimal transport and the Schr\"odinger problem where the transport cost is replaced by an entropic cost with a reference path measure. A dual Kantorovich type formulation and a Benamou-Brenier type representation formula of the entropic cost are derived, as well as contraction inequalities with respect to the entropic cost. This analogy is also illustrated with some numerical examples where the reference path measure is given by the Brownian or the Ornstein-Uhlenbeck process. Our point of view is measure theoretical and the relative entropy with respect to path measures plays a prominent role

Dynamical Optimal Transport on Discrete Surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows

Entropic Wasserstein Gradient Flows

This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e. Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically non-linear diffusion equations that model for instance porous media or crowd evolutions. These gradient flows define a suitable notion of weak solutions for these evolutions and they can be approximated in a stable way using discrete flows. These discrete flows are implicit Euler time stepping according to the Wasserstein metric. A bottleneck of these approaches is the high computational load induced by the resolution of each step. Indeed, this corresponds to the resolution of a convex optimization problem involving a Wasserstein distance to the previous iterate. Following several recent works on the approximation of Wasserstein distances, we consider a discrete flow induced by an entropic regularization of the transportation coupling. This entropic regularization allows one to trade the initial Wasserstein fidelity term for a Kulback-Leibler divergence, which is easier to deal with numerically. We show how KL proximal schemes, and in particular Dykstra's algorithm, can be used to compute each step of the regularized flow. The resulting algorithm is both fast, parallelizable and versatile, because it only requires multiplications by a Gibbs kernel. On Euclidean domains discretized on an uniform grid, this corresponds to a linear filtering (for instance a Gaussian filtering when $c$ is the squared Euclidean distance) which can be computed in nearly linear time. On more general domains, such as (possibly non-convex) shapes or on manifolds discretized by a triangular mesh, following a recently proposed numerical scheme for optimal transport, this Gibbs kernel multiplication is approximated by a short-time heat diffusion

Entropic and displacement interpolation: a computational approach using the Hilbert metric

Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities -- it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for interpolation of distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of recent developments in physics, probability theory, image processing, time-series analysis, and several other fields. In spite of extensive work and theoretical developments, the computation of OMT for large scale problems has remained a challenging task. An alternative framework for interpolating distributions, rooted in statistical mechanics and large deviations, is that of Schroedinger bridges (entropic interpolation). This may be seen as a stochastic regularization of OMT and can be cast as the stochastic control problem of steering the probability density of the state-vector of a dynamical system between two marginals. In this approach, however, the actual computation of flows had hardly received any attention. In recent work on Schroedinger bridges for Markov chains and quantum evolutions, we noted that the solution can be efficiently obtained from the fixed-point of a map which is contractive in the Hilbert metric. Thus, the purpose of this paper is to show that a similar approach can be taken in the context of diffusion processes which i) leads to a new proof of a classical result on Schroedinger bridges and ii) provides an efficient computational scheme for both, Schroedinger bridges and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.Comment: 20 pages, 7 figure

Convergence of Entropic Schemes for Optimal Transport and Gradient Flows

Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach has recently been used successfully to solve optimal transport related problems in several applied fields such as imaging sciences, machine learning and social sciences. The main reason for this success is that, in contrast to linear programming solvers, the resulting algorithms are highly parallelizable and take advantage of the geometry of the computational grid (e.g. an image or a triangulated mesh). The first contribution of this article is the proof of the $\Gamma$-convergence of the entropic regularized optimal transport problem towards the Monge-Kantorovich problem for the squared Euclidean norm cost function. This implies in particular the convergence of the optimal entropic regularized transport plan towards an optimal transport plan as the entropy vanishes. Optimal transport distances are also useful to define gradient flows as a limit of implicit Euler steps according to the transportation distance. Our second contribution is a proof that implicit steps according to the entropic regularized distance converge towards the original gradient flow when both the step size and the entropic penalty vanish (in some controlled way)