49 research outputs found
Ground Metric Learning on Graphs
Optimal transport (OT) distances between probability distributions are
parameterized by the ground metric they use between observations. Their
relevance for real-life applications strongly hinges on whether that ground
metric parameter is suitably chosen. Selecting it adaptively and
algorithmically from prior knowledge, the so-called ground metric learning GML)
problem, has therefore appeared in various settings. We consider it in this
paper when the learned metric is constrained to be a geodesic distance on a
graph that supports the measures of interest. This imposes a rich structure for
candidate metrics, but also enables far more efficient learning procedures when
compared to a direct optimization over the space of all metric matrices. We use
this setting to tackle an inverse problem stemming from the observation of a
density evolving with time: we seek a graph ground metric such that the OT
interpolation between the starting and ending densities that result from that
ground metric agrees with the observed evolution. This OT dynamic framework is
relevant to model natural phenomena exhibiting displacements of mass, such as
for instance the evolution of the color palette induced by the modification of
lighting and materials.Comment: Fixed sign of gradien
Half-quadratic transportation problems
We present a primal--dual memory efficient algorithm for solving a relaxed
version of the general transportation problem. Our approach approximates the
original cost function with a differentiable one that is solved as a sequence
of weighted quadratic transportation problems. The new formulation allows us to
solve differentiable, non-- convex transportation problems
Two-Stage Metric Learning
In this paper, we present a novel two-stage metric learning algorithm. We
first map each learning instance to a probability distribution by computing its
similarities to a set of fixed anchor points. Then, we define the distance in
the input data space as the Fisher information distance on the associated
statistical manifold. This induces in the input data space a new family of
distance metric with unique properties. Unlike kernelized metric learning, we
do not require the similarity measure to be positive semi-definite. Moreover,
it can also be interpreted as a local metric learning algorithm with well
defined distance approximation. We evaluate its performance on a number of
datasets. It outperforms significantly other metric learning methods and SVM.Comment: Accepted for publication in ICML 201
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
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