152 research outputs found
Variational Wasserstein Barycenters for Geometric Clustering
We propose to compute Wasserstein barycenters (WBs) by solving for Monge maps
with variational principle. We discuss the metric properties of WBs and explore
their connections, especially the connections of Monge WBs, to K-means
clustering and co-clustering. We also discuss the feasibility of Monge WBs on
unbalanced measures and spherical domains. We propose two new problems --
regularized K-means and Wasserstein barycenter compression. We demonstrate the
use of VWBs in solving these clustering-related problems
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
Regularized Wasserstein Means for Aligning Distributional Data
We propose to align distributional data from the perspective of Wasserstein
means. We raise the problem of regularizing Wasserstein means and propose
several terms tailored to tackle different problems. Our formulation is based
on the variational transportation to distribute a sparse discrete measure into
the target domain. The resulting sparse representation well captures the
desired property of the domain while reducing the mapping cost. We demonstrate
the scalability and robustness of our method with examples in domain
adaptation, point set registration, and skeleton layout
Sliced Wasserstein Distance for Learning Gaussian Mixture Models
Gaussian mixture models (GMM) are powerful parametric tools with many
applications in machine learning and computer vision. Expectation maximization
(EM) is the most popular algorithm for estimating the GMM parameters. However,
EM guarantees only convergence to a stationary point of the log-likelihood
function, which could be arbitrarily worse than the optimal solution. Inspired
by the relationship between the negative log-likelihood function and the
Kullback-Leibler (KL) divergence, we propose an alternative formulation for
estimating the GMM parameters using the sliced Wasserstein distance, which
gives rise to a new algorithm. Specifically, we propose minimizing the
sliced-Wasserstein distance between the mixture model and the data distribution
with respect to the GMM parameters. In contrast to the KL-divergence, the
energy landscape for the sliced-Wasserstein distance is more well-behaved and
therefore more suitable for a stochastic gradient descent scheme to obtain the
optimal GMM parameters. We show that our formulation results in parameter
estimates that are more robust to random initializations and demonstrate that
it can estimate high-dimensional data distributions more faithfully than the EM
algorithm
On the computation of Wasserstein barycenters
The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. In this paper, we state a general version of the equivalence of the Wasserstein barycenter problem to the n-coupling problem. As a consequence, the coupling to the sum principle (characterizing solutions to the n-coupling problem) provides a novel criterion for the explicit characterization of barycenters. Based on this criterion, we provide as a main contribution the simple to implement iterative swapping algorithm (ISA) for computing barycenters. The ISA is a completely non-parametric algorithm which provides a sharp image of the support of the barycenter and has a quadratic time complexity which is comparable to other well established algorithms designed to compute barycenters. The algorithm can also be applied to more complex optimization problems like the k-barycenter problem
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