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Distribution's template estimate with Wasserstein metrics
Emmanuel Boissard, Thibaut Le Gouic, Jean-Michel Loubes
TL;DR
For data with inner geometry or deformation, defining a meaningful mean distribution is difficult, and mean estimates based on parameterized deformations may lack established consistency in many cases. The paper extends Wasserstein barycenters to empirical measures and studies iterative barycenters; under suitable admissible deformations, it reports template estimation as replications increase.
Problem
For data with non-Euclidean geometry or deformation, the usual Euclidean mean may be inadequate, while defining a mean through squared distance raises existence and uniqueness questions tied to the data geometry.
Source: §1 · pp. 1–2Method
Results
For admissible deformations, the iterated barycenter corresponds to averaging the deformations and can coincide with the ordinary barycenter; the paper reports that the barycenter estimates the template as replications increase.
Source: §4.2 · p. 8, §4.3 · p. 9Takeaways & Limitations
In the stated warped-distribution setting, the barycenter provides an estimate of the template distribution as the number of replications grows.
Source: §4.3 · p. 9, §5.1 · p. 10The extended family of curves used for Wasserstein PCA is generally not geodesic over the full range of its parameter.
Source: §5.2 · p. 12
Abstract
from arXiv · showhide
In this paper we tackle the problem of comparing distributions of random variables and defining a mean pattern between a sample of random events. Using barycenters of measures in the Wasserstein space, we propose an iterative version as an estimation of the mean distribution. Moreover, when the distributions are a common measure warped by a centered random operator, then the barycenter enables to recover this distribution template.
1 Introduction
For high-dimensional data with non-Euclidean structure or deformation, defining a meaningful mean is difficult. The paper extends Wasserstein barycenters to empirical measures and studies their use in estimating a mean distribution and recovering a warped template.
1 Introduction
- Deformations and inner data geometry can make the Euclidean mean inadequate and the mean-behaviour problem difficult.The paper notes applications in functional data, image, and shape analysis.Source: §1 · p. 1
- Parameterized deformation models define mean behaviour by warping the data with mean parameters, but estimator consistency remains a theoretical issue in many cases.The introduction also identifies choosing an adequate distance as an alternative direction.Source: §1 · pp. 1–2
- Defining a mean by minimizing squared distance raises existence and uniqueness questions that depend on the geometry of the data space.Choosing an appropriate distance that reflects the information in the observations is itself difficult.Source: §1 · p. 2
- The paper extends Wasserstein barycenters to empirical measures, studies consistency of their estimate, and considers recovering a template from distributions warped by random operators.It also introduces an iterative barycenter estimate for the mean distribution.Source: §1 · p. 2
2 Barycenters in the Wasserstein space: Notations and general results
This section sets out Wasserstein barycenters and the transport notation used in the paper. It distinguishes these metric-space means from ordinary convex combinations of measures and recalls transport-map results for Euclidean space.
2 Barycenters in the Wasserstein space: Notations and general results
- Ordinary convex combinations of measures are not appropriate once the space of measures is endowed with a metric structure.For two absolutely continuous Euclidean measures, their barycenter is described using the Brenier map; the two-measure barycenter is an interpolant in McCann's sense.Source: §2 · pp. 2, 4
- The 2-Wasserstein distance is defined from quadratic transportation cost between probability measures with finite second moments.The cost uses couplings whose marginals are the two measures.Source: §2 · pp. 2–3
- A Wasserstein barycenter is the weighted Fréchet mean of measures, with existence in Euclidean space and uniqueness under suitable assumptions.One sufficient uniqueness condition is that one of the measures has a density with respect to Lebesgue measure.Source: §2 · p. 3
- For an absolutely continuous source measure in Euclidean space, Brenier's theorem gives an almost-everywhere unique optimal transport map, which is the gradient of a lower semi-continuous convex function.The map pushes the source measure to the target measure.Source: §2 · p. 4
3 Estimation of Barycenters of empirical measures
The section studies consistency of barycenters formed from approximations and introduces iterated barycenters as a distinct construction. It also notes computational properties that make iteration useful, including for warped-distribution settings.
3.1 Consistency of the approximated barycenter
- When approximating measures converge in Wasserstein distance, the sequence of their barycenters is compact and every limit point is a barycenter of the limiting measures.If any limiting measure is absolutely continuous with respect to Lebesgue measure, the barycenter is unique and the approximating barycenters converge to it.Source: §3.1 · p. 5
3.2 An Iterative version of barycenters of measures
- An iterated barycenter is formed by successively taking two-measure barycenters with appropriate weights, and it does not generally equal the ordinary barycenter.The paper identifies cases where the two notions match.Source: §3.2 · p. 5
- The iterated barycenter is consistent when the approximating measures converge in Wasserstein distance.The supplied theorem passage identifies this consistency result for the iterated construction.Source: §3.2 · p. 6
- Iteration yields a measure expressible through a linear combination of maps, which is useful for warping; in some cases its order independence allows parallel computation.The paper also notes that iterated barycenters are suited to computation using efficient numerical methods for McCann's interpolant.Source: §3.2 · p. 6
4 Deformations of a template measure
The section studies recovery of a template measure from push-forward-warped observations and identifies when Wasserstein barycenters or their iterative form can recover it. Its central estimation result requires centered deformations.
4 Deformations of a template measure
- The target is an original measure µ observed through push-forward deformations Tj#µ; prior curve-registration work also studied randomly warped versions of a distribution.The authors frame their study as an extension of that curve-registration setting.Source: §4 · p. 6
4.1 Admissible deformations
- Admissibility requires the relative maps Ti∘Tj^-1 to be gradients of convex functions.The paper denotes the set of gradients of convex functions by GCF(Ω).Source: §4.1 · pp. 6–7
- Proposition 4.1 lists admissible deformations including product continuous increasing maps, scale-location maps x ↦ ax+b (a>0, b∈R^n), and radial distortions with continuous increasing F satisfying F(0)=0.Source: §4.1 · p. 7
4.2 Barycenter of measures warped using admissible deformations
- For admissible deformations, the iterated barycenter corresponds to averaging the deformations and, in the stated case, coincides with the usual barycenter.The theorem considers µj=(Tj)#µ for µ absolutely continuous with respect to Lebesgue measure.Source: §4.2 · p. 8
- In dimension 1, all continuous increasing maps form an admissible deformation family, and the iterative estimator is the stated new feature relative to and.Earlier curve-registration work used quantile functions and showed convergence to (EH)#µ.Source: §4 · p. 6, §4.2 · p. 8
- In dimensions greater than 2, ordinary and iterated barycenters need not coincide; the Gaussian example has covariance matrices that generally differ.The paper presents centered non-degenerate Gaussian measures as a counterexample.Source: §4.2 · pp. 8–9
4.3 Template Estimation from admissible deformations
- With compactly supported µ and mean deformation ϕ=id, the equal-weight barycenter µB consistently estimates µ almost surely in Wasserstein distance as J→∞.The section also gives an error bound under the condition ∥T−id∥L2≤M a.s.Source: §4.3 · p. 9
- When the warping process is not centered, µ is not identifiable; the barycenter can estimate only the mean-deformed measure ϕ#µ.Here ϕ is the mean of the deformation process.Source: §4.3 · p. 9
5 Statistical Applications
Section 5.1 proposes iterative Wasserstein barycenters to estimate a mean distribution and shows template recovery under centered admissible deformations; it also illustrates examiner-score balancing. Section 5.2 develops Wasserstein PCA using curves from the barycenter and examines a translation example, while noting that the extended curves need not be geodesics throughout their parameter range.
5.1 Distribution Template estimation from empirical observations
- The paper uses the iterative barycenter of empirical distributions as a mean-shape estimate, identifying it as their Fréchet mean rather than pooling all observation points.The paper states that the pooled-point mean does not provide a good model of mean behaviour.Source: §5.1 · p. 10
- Under centered admissible deformations of a continuous template, the empirical iterative barycenter consistently estimates that template in W2 as the sample size grows.The result assumes observations are warped from an unknown template by a centered admissible deformation process.Source: §5.1 · p. 10
- Empirical B may exist without being unique; regularizing the empirical measures can ensure uniqueness, with other schemes allowed when they converge in Wasserstein distance.The convergence condition is stated for sample size n tending to infinity.Source: §5.1 · p. 11
- In the examiner application, pushing each group’s scores toward the barycenter yields a common ranking interpretable as scores from an average board.The setup assumes comparable candidate sampling across boards and aims to balance examiner differences.Source: §5.1 · p. 11
5.2 Principal Component Analysis with Wasserstein distance
- For clustering distributions, the paper expresses Wasserstein square distances through the transport maps from the barycenter.The cited passage gives the distance in terms of the squared L2(µB) norm of S_j − id.Source: §5.2 · p. 12
- Wasserstein PCA roots curves at the barycenter and seeks principal curves by minimizing summed squared distances from the distributions to those curves.The first component is computed from the barycenter, which serves as the mean of the distribution sample.Source: §5.2 · pp. 12–13
- The extended curve family is generally not geodesic over all t, and the paper’s translation example gives a first component by translating the barycenter along a constant shift.In that example, the barycenter density is f(x − θ̄) and the first component has density f(x − θ̄ − t).Source: §5.2 · pp. 12, 14
6 Appendix
The appendix establishes existence and convergence properties for Wasserstein barycenters and characterizes their relationship to solutions of a multimarginal problem.
6 Appendix
- The multimarginal problem (1) has a solution by a classical compactness argument.Source: §6 · p. 15
- A minimizer γ of (1) yields a barycenter ν = T#γ; conversely, a barycenter is represented by T#Πp for a projection Πp associated with an optimal coupling.The converse follows from the equality condition, which forces x = T(x1, ..., xJ) almost surely.Source: §6 · pp. 15–16
- Any limiting point of the sequence of barycenters is a minimizer for the barycenter problem associated with µ1, ..., µJ.The proof establishes compactness by obtaining tightness of the couplings and their push-forwards.Source: §6 · p. 16