79 research outputs found
Inverse optimal transport
Discrete optimal transportation problems arise in various contexts in
engineering, the sciences and the social sciences. Often the underlying cost
criterion is unknown, or only partly known, and the observed optimal solutions
are corrupted by noise. In this paper we propose a systematic approach to infer
unknown costs from noisy observations of optimal transportation plans. The
algorithm requires only the ability to solve the forward optimal transport
problem, which is a linear program, and to generate random numbers. It has a
Bayesian interpretation, and may also be viewed as a form of stochastic
optimization.
We illustrate the developed methodologies using the example of international
migration flows. Reported migration flow data captures (noisily) the number of
individuals moving from one country to another in a given period of time. It
can be interpreted as a noisy observation of an optimal transportation map,
with costs related to the geographical position of countries. We use a
graph-based formulation of the problem, with countries at the nodes of graphs
and non-zero weighted adjacencies only on edges between countries which share a
border. We use the proposed algorithm to estimate the weights, which represent
cost of transition, and to quantify uncertainty in these weights
Indirect Image Registration with Large Diffeomorphic Deformations
The paper adapts the large deformation diffeomorphic metric mapping framework
for image registration to the indirect setting where a template is registered
against a target that is given through indirect noisy observations. The
registration uses diffeomorphisms that transform the template through a (group)
action. These diffeomorphisms are generated by solving a flow equation that is
defined by a velocity field with certain regularity. The theoretical analysis
includes a proof that indirect image registration has solutions (existence)
that are stable and that converge as the data error tends so zero, so it
becomes a well-defined regularization method. The paper concludes with examples
of indirect image registration in 2D tomography with very sparse and/or highly
noisy data.Comment: 43 pages, 4 figures, 1 table; revise
Regularized Optimal Transport and the Rot Mover's Distance
This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification
Quadratically-Regularized Optimal Transport on Graphs
Optimal transportation provides a means of lifting distances between points
on a geometric domain to distances between signals over the domain, expressed
as probability distributions. On a graph, transportation problems can be used
to express challenging tasks involving matching supply to demand with minimal
shipment expense; in discrete language, these become minimum-cost network flow
problems. Regularization typically is needed to ensure uniqueness for the
linear ground distance case and to improve optimization convergence;
state-of-the-art techniques employ entropic regularization on the
transportation matrix. In this paper, we explore a quadratic alternative to
entropic regularization for transport over a graph. We theoretically analyze
the behavior of quadratically-regularized graph transport, characterizing how
regularization affects the structure of flows in the regime of small but
nonzero regularization. We further exploit elegant second-order structure in
the dual of this problem to derive an easily-implemented Newton-type
optimization algorithm.Comment: 27 page
Estimating ensemble flows on a hidden Markov chain
We propose a new framework to estimate the evolution of an ensemble of
indistinguishable agents on a hidden Markov chain using only aggregate output
data. This work can be viewed as an extension of the recent developments in
optimal mass transport and Schr\"odinger bridges to the finite state space
hidden Markov chain setting. The flow of the ensemble is estimated by solving a
maximum likelihood problem, which has a convex formulation at the
infinite-particle limit, and we develop a fast numerical algorithm for it. We
illustrate in two numerical examples how this framework can be used to track
the flow of identical and indistinguishable dynamical systems.Comment: 8 pages, 4 figure
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
- …