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A note on the computation of Wasserstein barycenters.∗
, 2015
"... We consider the problem of finding the barycenter of a finite set of probabilities on Rd with respect to the Wasserstein metric. We introduce an iterative procedure which consistenly approximates the barycenter under general conditions. These cover the case of probabilities in a locationscatter fa ..."
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We consider the problem of finding the barycenter of a finite set of probabilities on Rd with respect to the Wasserstein metric. We introduce an iterative procedure which consistenly approximates the barycenter under general conditions. These cover the case of probabilities in a locationscatter family, including the Gaussian case. The performance of the iterative procedure is illustrated through numerical simulations, which show fast convergence towards the barycenter.
Noname manuscript No. (will be inserted by the editor) Sliced and Radon Wasserstein Barycenters of Measures
"... the date of receipt and acceptance should be inserted later Abstract This article details two approaches to compute barycenters of measures using 1D Wasserstein distances along radial projections of the input measures. The first method makes use of the Radon transform of the measures, and the seco ..."
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the date of receipt and acceptance should be inserted later Abstract This article details two approaches to compute barycenters of measures using 1D Wasserstein distances along radial projections of the input measures. The first method makes use of the Radon transform of the measures, and the second is the solution of a convex optimization problem over the space of measures. We show several properties of these barycenters and explain their relationship. We show numerical approximation schemes based on a discrete Radon transform and on the resolution of a nonconvex optimization problem. We explore the respective merits and drawbacks of each approach on applications to two image processing problems: color transfer and texture mixing.
Wide consensus for parallelized inference.∗ Pedro C. ÁlvarezEsteban1
, 2015
"... We develop a general theory to address a consensusbased combination of estimations in a parallelized or distributed estimation setting. Taking into account the possibility of very discrepant estimations, instead of a full consensus we consider a “wide consensus ” procedure. The approach is based o ..."
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We develop a general theory to address a consensusbased combination of estimations in a parallelized or distributed estimation setting. Taking into account the possibility of very discrepant estimations, instead of a full consensus we consider a “wide consensus ” procedure. The approach is based on the consideration of trimmed barycenters in the Wasserstein space of probability distributions on Rd with finite second order moments. We include general existence and consistency results as well as characterizations of barycenters of probabilities that belong to (non necessarily elliptical) location and scatter familes. On these families, the effective computation of barycenters and distances can be addressed through a consistent iterative algorithm. Since, once a shape has been chosen, these computations just depend on the locations and scatters, the theory can be applied to cover with great generality a wide consensus approach for location and scatter estimation or for obtaining confidence regions.
Quantifying error in estimates of human brain fiber directions using Earth Mover’s Distance
"... Diffusionweighted MR imaging (DWI) is the only method we currently have to measure connections between different parts of the human brain in vivo. To elucidate the structure of these connections, algorithms for tracking bundles of axonal fibers through the subcortical white matter rely on local es ..."
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Diffusionweighted MR imaging (DWI) is the only method we currently have to measure connections between different parts of the human brain in vivo. To elucidate the structure of these connections, algorithms for tracking bundles of axonal fibers through the subcortical white matter rely on local estimates of the fiber orientation distribution function (fODF) in different parts of the brain. These functions describe the relative abundance of populations of axonal fibers crossing each other in each location. Multiple models exist for estimating fODFs. The quality of the resulting estimates can be quantified by means of a suitable measure of distance on the space of fODFs. However, there are multiple distance metrics that can be applied for this purpose, including smoothed Lp distances and the Wasserstein metrics. Here, we give four reasons for the use of the Earth Mover’s Distance (EMD) equipped with the arclength, as a distance metric. First, the EMD is an extension of the intuitive angular error metric, often used in the DWI literature. Second, the EMD is equally applicable to continuous fODFs or fODFs containing mixtures of Dirac deltas. Third, the EMD does not require specifying smoothing parameters. Finally, the EMD is useful in practice, as well as in simulations. This is because the error of an estimated fODF, as quantified by the EMD of this fODF from the ground truth is correlated with the replicate error: the EMD between the fODFs calculated on two repeated measurements. Though we cannot calculate the error of the estimate directly in experimental data measured in vivo (in contrast to simulation in which ground truth is known), we can use the replicate error, computed using repeated measurements, as a surrogate for the error. We demonstrate the application of computing the EMDbased replicate error in MRI data, creating anatomical contrast that is not observed with an estimate of model prediction error. 1
WASP: Scalable Bayes via barycenters of subset posteriors
"... The promise of Bayesian methods for big data sets has not fully been realized due to the lack of scalable computational algorithms. For massive data, it is necessary to store and process subsets on different machines in a distributed manner. We propose a simple, general, and highly efficient appr ..."
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The promise of Bayesian methods for big data sets has not fully been realized due to the lack of scalable computational algorithms. For massive data, it is necessary to store and process subsets on different machines in a distributed manner. We propose a simple, general, and highly efficient approach, which first runs a posterior sampling algorithm in parallel on different machines for subsets of a large data set. To combine these subset posteriors, we calculate the Wasserstein barycenter via a highly efficient linear program. The resulting estimate for the Wasserstein posterior (WASP) has an atomic form, facilitating straightforward estimation of posterior summaries of functionals of interest. The WASP approach allows posterior sampling algorithms for smaller data sets to be trivially scaled to huge data. We provide theoretical justification in terms of posterior consistency and algorithm efficiency. Examples are provided in complex settings including Gaussian process regression and nonparametric Bayes mixture models. 1
Transport between RGB Images Motivated by Dynamic Optimal Transport
, 2016
"... We propose two models for the interpolation between RGB images based on the dynamic optimal transport model of Benamou and Brenier [8]. While the application of dynamic optimal transport and its extensions to unbalanced transform were examined for grayvalues images in various papers, this is the fi ..."
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We propose two models for the interpolation between RGB images based on the dynamic optimal transport model of Benamou and Brenier [8]. While the application of dynamic optimal transport and its extensions to unbalanced transform were examined for grayvalues images in various papers, this is the first attempt to generalize the idea to color images. The nontrivial task to incorporate color into the model is tackled by considering RGB images as threedimensional arrays, where the transport in the RGB direction is performed in a periodic way. Following the approach of Papadakis et al. [35] for grayvalue images we propose two discrete variational models, a constrained and a penalized one which can also handle unbalanced transport. We show that a minimizer of our discrete model exists, but it is not unique for some special initial/final images. For minimizing the resulting functionals we apply a primaldual algorithm. One step of this algorithm requires the solution of a fourdimensional discretized Poisson equation with various boundary conditions in each dimension. For instance, for the penalized approach we have simultaneously zero, mirror and periodic boundary conditions. The solution can be computed efficiently using fast SinI, CosII and Fourier transforms. Numerical examples demonstrate the meaningfulness of our model. 1.