Regularized Barycenters in the Wasserstein Space

07/11/2017
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This paper is an overview of results that have been obtain in [2] on the convex regularization of Wasserstein barycenters for random measures supported on Rd. We discuss the existence and uniqueness of such barycenters for a large class of regularizing functions. A stability result of regularized barycenters in terms of Bregman distance associated to the convex regularization term is also given. Additionally we discuss the convergence of the regularized empirical barycenter of a set of n iid random probability measures towards its population counterpart in the real line case, and we discuss its rate of convergence. This approach is shown to be appropriate for the statistical analysis of discrete or absolutely continuous random measures. In this setting, we propose an efficient minimization algorithm based on accelerated gradient descent for the computation of regularized Wasserstein barycenters.

Regularized Barycenters in the Wasserstein Space

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application/pdf Regularized Barycenters in the Wasserstein Space Elsa Cazelles, Jérémie Bigot, Nicolas Papadakis
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This paper is an overview of results that have been obtain in [2] on the convex regularization of Wasserstein barycenters for random measures supported on Rd. We discuss the existence and uniqueness of such barycenters for a large class of regularizing functions. A stability result of regularized barycenters in terms of Bregman distance associated to the convex regularization term is also given. Additionally we discuss the convergence of the regularized empirical barycenter of a set of n iid random probability measures towards its population counterpart in the real line case, and we discuss its rate of convergence. This approach is shown to be appropriate for the statistical analysis of discrete or absolutely continuous random measures. In this setting, we propose an efficient minimization algorithm based on accelerated gradient descent for the computation of regularized Wasserstein barycenters.
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Regularized Barycenters in the Wasserstein Space Elsa Cazelles, Jérémie Bigot, and Nicolas Papadakis Université de Bordeaux, CNRS, Institut de Mathématiques de Bordeaux, UMR 5251, elsa.cazelles@u-bordeaux.fr Abstract. This paper is an overview of results that have been obtain in [2] on the convex regularization of Wasserstein barycenters for random measures supported on Rd . We discuss the existence and uniqueness of such barycenters for a large class of regularizing functions. A stability result of regularized barycenters in terms of Bregman distance associated to the convex regularization term is also given. Additionally we discuss the convergence of the regularized empirical barycenter of a set of n iid random probability measures towards its population counterpart in the real line case, and we discuss its rate of convergence. This approach is shown to be appropriate for the statistical analysis of discrete or ab- solutely continuous random measures. In this setting, we propose an efficient minimization algorithm based on accelerated gradient descent for the computation of regularized Wasserstein barycenters. Keywords: Wasserstein space, Fréchet mean, Barycenter of probability measures, Convex regularization, Bregman divergence 1 Introduction This paper is concerned by the statistical analysis of data sets whose elements may be modeled as random probability measures supported on Rd . It is an overview of results that have been obtain in [2]. In the special case of one dimen- sion (d = 1), we are able to provide refined results on the study of a sequence of discrete measures or probability density functions (e.g. histograms) that can be viewed as random probability measures. Such data sets appear in various research fields. Examples can be found in neuroscience [10], biodemographic and genomics studies [11], economics [7], as well as in biomedical imaging [9]. In this paper, we focus on first-order statistics methods for the purpose of estimating, from such data, a population mean measure or density function. The notion of averaging depends on the metric that is chosen to compare elements in a given data set. In this work, we consider the Wasserstein distance W2 associated to the quadratic cost for the comparison of probability measures. Let Ω be a subset of Rd and P2(Ω) be the set of probability measures supported on Ω with finite order second moment. Definition 1. As introduced in [1], an empirical Wasserstein barycenter ν̄n of a set of n probability measures ν1, . . . , νn (not necessarily random) in P2(Ω) is defined as a minimizer of µ 7→ 1 n n X i=1 W2 2 (µ, νi), over µ ∈ P2(Ω). (1) The Wasserstein barycenter corresponds to the notion of empirical Fréchet mean [6] that is an extension of the usual Euclidean barycenter to nonlinear metric spaces. However, depending on the data at hand, such a barycenter may be irregu- lar. As an example let us consider a real data set of neural spike trains which is publicly available from the MBI website1 . During a squared-path task, the spiking activity of a movement-encoded neuron of a monkey has been recorded during 5 seconds over n = 60 repeated trials. Each spike train is then smoothed using a Gaussian kernel (further details on the data collection can be found in [10]). For each trial 1 ≤ i ≤ n, we let νi be the measure with probability density function (pdf) proportional to the sum of these Gaussian kernels centered at the times of spikes. The resulting data are displayed in Fig. 1(a). For probability measures supported on the real line, computing a Wasserstein barycenter simply amounts to averaging the quantile functions of the νi’s (see e.g. Section 6.1 in [1]). The pdf of the Wasserstein barycenter ν̄n is displayed in Fig. 1(b). This ap- proach clearly leads to the estimation of a very irregular mean template density of spiking activity. In this paper, we thus introduce a convex regularization of the optimization problem (1) for the purpose of obtaining a regularized Wasserstein barycenter. In this way, by choosing an appropriate regularizing function (e.g. the negative entropy in Subsection 2.1), it is of possible to enforce this barycenter to be absolutely continuous with respect to the Lebesgue measure on Rd . 2 Regularization of barycenters We choose to add a penalty directly into the computation of the Wasserstein barycenter in order to smooth the Fréchet mean and to remove the influence of noise in the data. Definition 2. Let Pν n = 1 n Pn i=1 δνi where δνi is the dirac distribution at νi. We define a regularized empirical barycenter µγ Pν n of the discrete measure Pν n as a minimizer of µ 7→ 1 n n X i=1 W2 2 (µ, νi) + γE(µ) over µ ∈ P2(Ω), (2) where P2(Ω) is the space of probability measures on Ω with finite second order moment, E : P2(Ω) → R+ is a smooth convex penalty function, and γ > 0 is a regularization parameter. In what follows, we present the main properties on the regularized empirical Wasserstein barycenter µγ Pν n . 1 http://mbi.osu.edu/2012/stwdescription.html 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.02 0.04 0.06 0.08 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.02 0.04 0.06 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.02 0.04 0.06 (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 (b) Fig. 1. (a) A subset of 3 smoothed neural spike trains out of n = 60. Each row rep- resents one trial and the pdf obtained by smoothing each spike train with a Gaus- sian kernel of width 50 milliseconds. (b) Probability density function of the empirical Wasserstein barycenter ν̄n for this data set. 2.1 Existence and uniqueness We consider the wider problem of min µ∈P2(Ω) Jγ P (µ) = Z W2 2 (µ, ν)dP(ν) + γE(µ). (3) Hence, (2) corresponds to the minimization problem (3) where P is discrete ie P