33 research outputs found
Stochastic Wasserstein Barycenters
We present a stochastic algorithm to compute the barycenter of a set of
probability distributions under the Wasserstein metric from optimal transport.
Unlike previous approaches, our method extends to continuous input
distributions and allows the support of the barycenter to be adjusted in each
iteration. We tackle the problem without regularization, allowing us to recover
a sharp output whose support is contained within the support of the true
barycenter. We give examples where our algorithm recovers a more meaningful
barycenter than previous work. Our method is versatile and can be extended to
applications such as generating super samples from a given distribution and
recovering blue noise approximations.Comment: ICML 201
Recommended from our members
Computational Optimal Transport
Optimal transport is the mathematical discipline of matching supply to demand while minimizing shipping costs. This matching problem becomes extremely challenging as the quantity of supply and demand points increases; modern applications must cope with thousands or millions of these at a time. Here, we introduce the computational optimal transport problem and summarize recent ideas for achieving new heights in efficiency and scalability
Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm
Starting from Brenier's relaxed formulation of the incompressible Euler
equation in terms of geodesics in the group of measure-preserving
diffeomorphisms, we propose a numerical method based on Sinkhorn's algorithm
for the entropic regularization of optimal transport. We also make a detailed
comparison of this entropic regularization with the so-called Bredinger
entropic interpolation problem. Numerical results in dimension one and two
illustrate the feasibility of the method
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
Second order models for optimal transport and cubic splines on the Wasserstein space
On the space of probability densities, we extend the Wasserstein geodesics to
the case of higher-order interpolation such as cubic spline interpolation.
After presenting the natural extension of cubic splines to the Wasserstein
space, we propose a simpler approach based on the relaxation of the variational
problem on the path space. We explore two different numerical approaches, one
based on multi-marginal optimal transport and entropic regularization and the
other based on semi-discrete optimal transport
Efficient and Accurate Optimal Transport with Mirror Descent and Conjugate Gradients
We design a novel algorithm for optimal transport by drawing from the
entropic optimal transport, mirror descent and conjugate gradients literatures.
Our scalable and GPU parallelizable algorithm is able to compute the
Wasserstein distance with extreme precision, reaching relative error rates of
without numerical stability issues. Empirically, the algorithm
converges to high precision solutions more quickly in terms of wall-clock time
than a variety of algorithms including log-domain stabilized Sinkhorn's
Algorithm. We provide careful ablations with respect to algorithm and problem
parameters, and present benchmarking over upsampled MNIST images, comparing to
various recent algorithms over high-dimensional problems. The results suggest
that our algorithm can be a useful addition to the practitioner's optimal
transport toolkit
Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm
We present a novel algorithm to estimate the barycenter of arbitrary
probability distributions with respect to the Sinkhorn divergence. Based on a
Frank-Wolfe optimization strategy, our approach proceeds by populating the
support of the barycenter incrementally, without requiring any pre-allocation.
We consider discrete as well as continuous distributions, proving convergence
rates of the proposed algorithm in both settings. Key elements of our analysis
are a new result showing that the Sinkhorn divergence on compact domains has
Lipschitz continuous gradient with respect to the Total Variation and a
characterization of the sample complexity of Sinkhorn potentials. Experiments
validate the effectiveness of our method in practice.Comment: 46 pages, 8 figure