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On the $q$-integrability of $p$-Wasserstein barycenters

Camilla Brizzi, Lorenzo Portinale

Abstract

We study the $L^q$-regularity of the density of barycenters of $N$ probability measures on $\mathbb{R}^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). According to a previous result by the first author and collaborators, if one marginal is absolutely continuous, so is the $W_p$-barycenter. The next natural question is whether the $L^q$- regularity on the marginals is also preserved for any $q > 1$, as in the classical case ($p=2$) of Agueh--Carlier, or for $W_p$-geodesics ($N=2$). Here we prove that this is the case if one marginal belongs to $L^q$ and the supports of all the marginals satisfy suitable geometric assumptions. However, we show that, as soon as $N>2$, it is possible to find examples of $W_p$-barycenters which are not $q$-integrable, even if one marginal is compactly supported and bounded, thus highlighting the role played by the geometry of the supports. Furthermore, we provide a general estimate of the $L^q$-norm, including a detailed study of the sources of singularities, and a characterization of the $W_p$-barycenters à la Agueh--Carlier in terms of the associated Kantorovich potentials. Finally, we explicitly compute the $W_p$-barycenters of measures obtained as push-forward of special affine transformations. In this case, regularity holds without any additional requirement on the supports.

On the $q$-integrability of $p$-Wasserstein barycenters

Abstract

We study the -regularity of the density of barycenters of probability measures on with respect to the -Wasserstein metric (). According to a previous result by the first author and collaborators, if one marginal is absolutely continuous, so is the -barycenter. The next natural question is whether the - regularity on the marginals is also preserved for any , as in the classical case () of Agueh--Carlier, or for -geodesics (). Here we prove that this is the case if one marginal belongs to and the supports of all the marginals satisfy suitable geometric assumptions. However, we show that, as soon as , it is possible to find examples of -barycenters which are not -integrable, even if one marginal is compactly supported and bounded, thus highlighting the role played by the geometry of the supports. Furthermore, we provide a general estimate of the -norm, including a detailed study of the sources of singularities, and a characterization of the -barycenters à la Agueh--Carlier in terms of the associated Kantorovich potentials. Finally, we explicitly compute the -barycenters of measures obtained as push-forward of special affine transformations. In this case, regularity holds without any additional requirement on the supports.
Paper Structure (10 sections, 9 theorems, 117 equations)

This paper contains 10 sections, 9 theorems, 117 equations.

Table of Contents

  1. $L^q$-regularity (and counterexamples) of the $p$-Wasserstein barycenter
  2. Estimates, integrability, and counterexamples for concentrated measures
  3. $L^q$-estimates for general measures with distant support
  4. Integrability for $p\geq 2$
  5. Integrability for $p\in(1,2)$
  6. A general estimate on the $L^q$-norm of $p$-Wasserstein barycenters
  7. Optimal maps for $p$-Wasserstein distance and barycenters via affine transformations
  8. Optimality of affine transformations for the $p$-Wasserstein distance
  9. Barycenters of measures under affine transformations
  10. Injectivity estimate and proof of Theorem \ref{['th:absolutecontinuity']}

Key Result

Proposition 1.1

Let $\mu_1 = f_1 \mathrm{d} \mathcal{L}^d$ such that eq:compact_supports and with $f_1 \in L^q$. Assume Then $g_p \in L^q$ and we have where $C \in \mathbb{R}_+$ depends on $M$, $p$, $d$, and - respectively - on

Theorems & Definitions (27)

  • Proposition 1.1: Discrete integrability
  • proof
  • Remark 1.2
  • Example 1.3: Counterxamples to integrability
  • proof : Proof of Theorem \ref{['thm:integrability_distant_intro']} $(p \geq 2)$
  • proof : Proof of Theorem \ref{['thm:integrability_distant_intro']} $(p \in (1,2))$
  • Proposition 2.1: Differentiability of $\mathrm{bar}_p$
  • proof
  • Remark 2.2: $D_S^1$ with $S \in \mathcal{F}_1$
  • Corollary 2.3: Local regularity of $\mathrm{bar}_p$
  • ...and 17 more