37 research outputs found
Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm
We present a novel algorithm to estimate the barycenter of arbitrary
probability distributions with respect to the Sinkhorn divergence. Based on a
Frank-Wolfe optimization strategy, our approach proceeds by populating the
support of the barycenter incrementally, without requiring any pre-allocation.
We consider discrete as well as continuous distributions, proving convergence
rates of the proposed algorithm in both settings. Key elements of our analysis
are a new result showing that the Sinkhorn divergence on compact domains has
Lipschitz continuous gradient with respect to the Total Variation and a
characterization of the sample complexity of Sinkhorn potentials. Experiments
validate the effectiveness of our method in practice.Comment: 46 pages, 8 figure
Entropic Optimal Transport in Machine Learning: applications to distributional regression, barycentric estimation and probability matching
Regularised optimal transport theory has been gaining increasing interest in machine learning as a versatile tool to handle and compare probability measures. Entropy-based regularisations, known as Sinkhorn divergences, have proved successful in a wide range of applications: as a metric for clustering and barycenters estimation, as a tool to transfer information in domain adaptation, and as a fitting loss for generative models, to name a few. Given this success, it is crucial to investigate the statistical and optimization properties of such models. These aspects are instrumental to design new and principled paradigms that contribute to further advance the field. Nonetheless, questions on asymptotic guarantees of the estimators based on Entropic Optimal Transport have received less attention. In this thesis we target such questions, focusing on three major settings where Entropic Optimal Transport has been used: learning histograms in supervised frameworks, barycenter estimation and probability matching. We present the first consistent estimator for learning with Sinkhorn loss in supervised settings, with explicit excess risk bounds. We propose a novel algorithm for Sinkhorn barycenters that handles arbitrary probability distributions with provable global convergence guarantees. Finally, we address generative models with Sinkhorn divergence as loss function: we analyse the role of the latent distribution and the generator from a modelling and statistical perspective. We propose a method that learns the latent distribution and the generator jointly and we characterize the generalization properties of such estimator. Overall, the tools developed in this work contribute to the understanding of the theoretical properties of Entropic Optimal Transport and their versatility in machine learning
Generalized conditional gradient: analysis of convergence and applications
The objectives of this technical report is to provide additional results on
the generalized conditional gradient methods introduced by Bredies et al.
[BLM05]. Indeed , when the objective function is smooth, we provide a novel
certificate of optimality and we show that the algorithm has a linear
convergence rate. Applications of this algorithm are also discussed
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
Debiased Sinkhorn barycenters
International audienceEntropy regularization in optimal transport (OT) has been the driver of many recent interests for Wasserstein metrics and barycenters in machine learning. It allows to keep the appealing geometrical properties of the unregularized Wasserstein distance while having a significantly lower complexity thanks to Sinkhorn's algorithm. However, entropy brings some inherent smoothing bias, resulting for example in blurred barycenters. This side effect has prompted an increasing temptation in the community to settle for a slower algorithm such as log-domain stabilized Sinkhorn which breaks the parallel structure that can be leveraged on GPUs, or even go back to unregularized OT. Here we show how this bias is tightly linked to the reference measure that defines the entropy regularizer and propose debiased Wasserstein barycenters that preserve the best of both worlds: fast Sinkhorn-like iterations without entropy smoothing. Theoretically, we prove that the entropic OT barycenter of univariate Gaussians is a Gaussian and quantify its variance bias. This result is obtained by extending the differentiability and convexity of entropic OT to sub-Gaussian measures with unbounded supports. Empirically, we illustrate the reduced blurring and the computational advantage on various applications