2,109 research outputs found
A Smoothed Dual Approach for Variational Wasserstein Problems
Variational problems that involve Wasserstein distances have been recently
proposed to summarize and learn from probability measures. Despite being
conceptually simple, such problems are computationally challenging because they
involve minimizing over quantities (Wasserstein distances) that are themselves
hard to compute. We show that the dual formulation of Wasserstein variational
problems introduced recently by Carlier et al. (2014) can be regularized using
an entropic smoothing, which leads to smooth, differentiable, convex
optimization problems that are simpler to implement and numerically more
stable. We illustrate the versatility of this approach by applying it to the
computation of Wasserstein barycenters and gradient flows of spacial
regularization functionals
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 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
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 -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)
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
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