9 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
Functional optimal transport: map estimation and domain adaptation for functional data
We introduce a formulation of optimal transport problem for distributions on
function spaces, where the stochastic map between functional domains can be
partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt
operator mapping a Hilbert space of functions to another. For numerous machine
learning tasks, data can be naturally viewed as samples drawn from spaces of
functions, such as curves and surfaces, in high dimensions. Optimal transport
for functional data analysis provides a useful framework of treatment for such
domains. In this work, we develop an efficient algorithm for finding the
stochastic transport map between functional domains and provide theoretical
guarantees on the existence, uniqueness, and consistency of our estimate for
the Hilbert-Schmidt operator. We validate our method on synthetic datasets and
study the geometric properties of the transport map. Experiments on real-world
datasets of robot arm trajectories further demonstrate the effectiveness of our
method on applications in domain adaptation.Comment: 23 pages, 6 figures, 2 table
BoMb-OT: On Batch of Mini-batches Optimal Transport
Mini-batch optimal transport (m-OT) has been successfully used in practical
applications that involve probability measures with intractable density, or
probability measures with a very high number of supports. The m-OT solves
several sparser optimal transport problems and then returns the average of
their costs and transportation plans. Despite its scalability advantage, the
m-OT does not consider the relationship between mini-batches which leads to
undesirable estimation. Moreover, the m-OT does not approximate a proper metric
between probability measures since the identity property is not satisfied. To
address these problems, we propose a novel mini-batching scheme for optimal
transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that finds
the optimal coupling between mini-batches and it can be seen as an
approximation to a well-defined distance on the space of probability measures.
Furthermore, we show that the m-OT is a limit of the entropic regularized
version of the BoMb-OT when the regularized parameter goes to infinity.
Finally, we carry out extensive experiments to show that the BoMb-OT can
estimate a better transportation plan between two original measures than the
m-OT. It leads to a favorable performance of the BoMb-OT in the matching and
color transfer tasks. Furthermore, we observe that the BoMb-OT also provides a
better objective loss than the m-OT for doing approximate Bayesian computation,
estimating parameters of interest in parametric generative models, and learning
non-parametric generative models with gradient flow.Comment: 36 pages, 20 figure