10 research outputs found
Sinkhorn Divergences for Unbalanced Optimal Transport
Optimal transport induces the Earth Mover's (Wasserstein) distance between
probability distributions, a geometric divergence that is relevant to a wide
range of problems. Over the last decade, two relaxations of optimal transport
have been studied in depth: unbalanced transport, which is robust to the
presence of outliers and can be used when distributions don't have the same
total mass; entropy-regularized transport, which is robust to sampling noise
and lends itself to fast computations using the Sinkhorn algorithm. This paper
combines both lines of work to put robust optimal transport on solid ground.
Our main contribution is a generalization of the Sinkhorn algorithm to
unbalanced transport: our method alternates between the standard Sinkhorn
updates and the pointwise application of a contractive function. This implies
that entropic transport solvers on grid images, point clouds and sampled
distributions can all be modified easily to support unbalanced transport, with
a proof of linear convergence that holds in all settings. We then show how to
use this method to define pseudo-distances on the full space of positive
measures that satisfy key geometric axioms: (unbalanced) Sinkhorn divergences
are differentiable, positive, definite, convex, statistically robust and avoid
any "entropic bias" towards a shrinkage of the measures' supports
The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation
Comparing metric measure spaces (i.e. a metric space endowed with
aprobability distribution) is at the heart of many machine learning problems.
The most popular distance between such metric measure spaces is
theGromov-Wasserstein (GW) distance, which is the solution of a quadratic
assignment problem. The GW distance is however limited to the comparison of
metric measure spaces endowed with a probability distribution.To alleviate this
issue, we introduce two Unbalanced Gromov-Wasserstein formulations: a distance
and a more tractable upper-bounding relaxation.They both allow the comparison
of metric spaces equipped with arbitrary positive measures up to isometries.
The first formulation is a positive and definite divergence based on a
relaxation of the mass conservation constraint using a novel type of
quadratically-homogeneous divergence. This divergence works hand in hand with
the entropic regularization approach which is popular to solve large scale
optimal transport problems. We show that the underlying non-convex optimization
problem can be efficiently tackled using a highly parallelizable and
GPU-friendly iterative scheme. The second formulation is a distance between
mm-spaces up to isometries based on a conic lifting. Lastly, we provide
numerical experiments onsynthetic examples and domain adaptation data with a
Positive-Unlabeled learning task to highlight the salient features of the
unbalanced divergence and its potential applications in ML
Unbalanced Multi-Marginal Optimal Transport
Entropy regularized optimal transport and its multi-marginal generalization
have attracted increasing attention in various applications, in particular due
to efficient Sinkhorn-like algorithms for computing optimal transport plans.
However, it is often desirable that the marginals of the optimal transport plan
do not match the given measures exactly, which led to the introduction of the
so-called unbalanced optimal transport. Since unbalanced methods were not
examined for the multi-marginal setting so far, we address this topic in the
present paper. More precisely, we introduce the unbalanced multi-marginal
optimal transport problem and its dual, and show that a unique optimal
transport plan exists under mild assumptions. Further, we generalize the
Sinkhorn algorithm for regularized unbalanced optimal transport to the
multi-marginal setting and prove its convergence. If the cost function
decouples according to a tree, the iterates can be computed efficiently. At the
end, we discuss three applications of our framework, namely two barycenter
problems and a transfer operator approach, where we establish a relation
between the barycenter problem and the multi-marginal optimal transport with an
appropriate tree-structured cost function
Transfer Operators from Optimal Transport Plans for Coherent Set Detection
The topic of this study lies in the intersection of two fields. One is
related with analyzing transport phenomena in complicated flows.For this
purpose, we use so-called coherent sets: non-dispersing, possibly moving
regions in the flow's domain. The other is concerned with reconstructing a flow
field from observing its action on a measure, which we address by optimal
transport. We show that the framework of optimal transport is well suited for
delivering the formal requirements on which a coherent-set analysis can be
based on. The necessary noise-robustness requirement of coherence can be
matched by the computationally efficient concept of unbalanced regularized
optimal transport. Moreover, the applied regularization can be interpreted as
an optimal way of retrieving the full dynamics given the extremely restricted
information of an initial and a final distribution of particles moving
according to Brownian motion
Entropy-Transport distances between unbalanced metric measure spaces
Inspired by the recent theory of Entropy-Transport problems and by the
-distance of Sturm on normalised metric measure spaces, we define a
new class of complete and separable distances between metric measure spaces of
possibly different total mass.
We provide several explicit examples of such distances, where a prominent
role is played by a geodesic metric based on the Hellinger-Kantorovich
distance. Moreover, we discuss some limiting cases of the theory, recovering
the "pure transport" -distance and introducing a new class of "pure
entropic" distances.
We also study in detail the topology induced by such Entropy-Transport
metrics, showing some compactness and stability results for metric measure
spaces satisfying Ricci curvature lower bounds in a synthetic sense.Comment: 36 pages. Comments are welcome
Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form
International audienceAlthough optimal transport (OT) problems admit closed form solutions in a very few notable cases, e.g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry. On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms. In this paper, we propose to fill the void at the intersection between these two schools of thought in OT by proving that the entropy-regularized optimal transport problem between two Gaussian measures admits a closed form. Contrary to the unregularized case, for which the explicit form is given by the Wasserstein-Bures distance, the closed form we obtain is differentiable everywhere, even for Gaussians with degenerate covariance matrices. We obtain this closed form solution by solving the fixed-point equation behind Sinkhorn's algorithm, the default method for computing entropic regularized OT. Remarkably, this approach extends to the generalized unbalanced case-where Gaussian measures are scaled by positive constants. This extension leads to a closed form expression for unbalanced Gaussians as well, and highlights the mass transportation / destruction trade-off seen in unbalanced optimal transport. Moreover, in both settings, we show that the optimal transportation plans are (scaled) Gaussians and provide analytical formulas of their parameters. These formulas constitute the first non-trivial closed forms for entropy-regularized optimal transport, thus providing a ground truth for the analysis of entropic OT and Sinkhorn's algorithm