A canonical barycenter via Wasserstein regularization

TL;DR

This paper introduces a new notion of barycenter in metric measure spaces using Wasserstein regularization, providing a well-defined, unique measure supported on classical barycenter points under certain conditions.

Contribution

It proposes a novel Wasserstein regularization approach to define a canonical barycenter in metric measure spaces, ensuring uniqueness and support on classical barycenter points.

Findings

The barycenter $B()$ is well-defined and supported on classical barycenter points.

Regularization with Wasserstein distance yields a unique measure.

Various properties of the barycenter are analyzed.

Abstract

We introduce a weak notion of barycenter of a probability measure $\mu$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(\mu)$ is well defined; it is a probability measure on $X$ supported on the set of the usual metric barycenter points of the given measure $\mu$. The definition uses the canonical embedding of the metric space $X$ into its Wasserstein space $P(X)$, pushing a given measure $\mu$ forward to a measure on $P(X)$. We then regularize the measure by the Wasserstein distance to the reference measure ${\bf m}$, and obtain a uniquely defined measure on $X$ supported on the barycentric points of $\mu$. We investigate various properties of $B(\mu)$

Equations

$$\displaystyle b(\mu)={\rm argmin}\Big{(}y\mapsto\int_{X}d^{2}(x,y)d\mu(x)\Big{)}.$$

$$W_2(\mu ,\nu ):=\inf {\int }_{X\times X}{d}^2(x,y)d\gamma (x,y)$$

$$\nu \mapsto {\int }_X{W}_2^2(\nu ,{\delta }_x)d\mu (x)$$

$$F_{ϵ}(\nu ):={\int }_X{\int }_X{d}^2(y,x)d\mu (x)d\nu (y)+ϵW_2^2(\mathbf{m},\nu ).$$

$$\mathrm{supp}(\mathbf{m})\cap b(\mu )\subseteq \mathrm{supp}B(\mu )\subseteq b(\mu ),$$

$$\{B(\mu )\}={\mathrm{argmin}}_{\mathrm{supp}\nu \subset b(\mu )}{W}_2(\nu ,\mathbf{m}).$$

$$\displaystyle d_{0}=\min\Big{(}y\mapsto\int_{X}d^{2}(x,y)d\mu(x)\Big{)}.\quad$$

$${\int }_X{d}^2(x,y)d\mu (x)>d_0+\eta for\; ally\in B_r(z).$$

$${\int }_X{\int }_X{d}^2(x,y)d\mu (x)d{\mu }_0(y)$$

$$=d_0+\eta {\mu }_0(B_r(z)).$$

$$F_{ϵ}({\mu }_{ϵ})\le F_{ϵ}(\tilde{\mu })=d_0+ϵW_2^2(\mathbf{m},\tilde{\mu })$$

$$\le {\int }_X{\int }_X{d}^2(x,y)d\mu (x)d{\mu }_0(y)-\eta {\mu }_0(B_r(z))+ϵW_2^2(\mathbf{m},{\mu }_0)$$

$$+ϵ\left(W_2^2(\mathbf{m},\tilde{\mu })-W_2^2(\mathbf{m},{\mu }_0)\right)$$

$$\le F_{ϵ}({\mu }_0)-\frac{\eta }{2}{\mu }_0(B_r(z))for\; sufficiently\; smallϵ.$$

$$W_2^2(\bar{\mu },\mathbf{m})\le W_2^2({\mu }_0,\mathbf{m})-\delta .$$

$$W_2^2(\mathbf{m},{\mu }_0)-W_2^2(\mathbf{m},{\mu }_{ϵ})$$

$$F_{ϵ}({\mu }_{ϵ})$$

$$\tilde{\mu }=\bar{\mu }-{\pi }_{\#}^2{\gamma }_{B_r(\bar{x})\times X}+\mathbf{m}(B_r(\bar{x})){\delta }_{\bar{x}}$$

$$\tilde{\gamma }=\gamma -{\gamma }_{B_r(\bar{x})\times X}+{\mathbf{m}}_{B_r(\bar{x})}\otimes {\delta }_{\bar{x}}.$$

$${\int }_{B_r(\bar{x})\times X}{d}^2(x,y)d\gamma (x,y)>4\mathbf{m}(B_r(\bar{x}))r^2.$$

$$\int d^2(x,y)d{\mathbf{m}}_{B_r(x)}\otimes {\delta }_{\bar{x}}(x,y)\le r^2\mathbf{m}(B_r(\bar{x})),$$

$$W_2^2(\mathbf{m},\tilde{\mu })\le W_2^2(\mathbf{m},\bar{\mu })-3\mathbf{m}(B_r(\bar{x}))r^2,$$

$$T_{\#}\mathbf{m}=B(\mu ).$$

$$\nabla f(x)={\nabla }_x(d^2(x,y)).$$

$$W_2(\mathbf{m},T_{\#}\mathbf{m})\le {\int }_X{d}^2(x,T(x))d\mathbf{m}(x)\le {\int }_X{d}^2(x,T_{\nu }(x))d\mathbf{m}(x)=W_2^2(\mathbf{m},\nu )$$

$$\mathrm{var}(B(\mu ))\le \mathrm{var}(\mu ).$$

$$\mathrm{var}(B(\mu ))$$

$$\mathrm{supp}(B(B(\mu )))=\mathrm{supp}(B^2(\mu ))=\mathrm{supp}(\mu )=\mathrm{supp}(B(B^{N-1}(\mu )))$$

$$\mu =B^N(\mu )=B(B^{N-1}(\mu ))=B(B(\mu ))=B^2(\mu ).$$

$${\int }_X\varphi (x)d(B(\mu ))(x)\le {\int }_X\varphi (x)d\mu (x).$$

$$\varphi (y)\le {\int }_X\varphi (x)d\mu (x).$$

$$x={\int }_{{\mathbb{R}}^n}yd{\pi }_x(y)$$

Full text

1 Introduction

In this paper, we propose a canonical weak notion of barycenter of a probability measure on a compact metric measure space $(X,d,{\bf m})$, where we assume that ${\bf m}$ is a probability measure.

For a separable compact metric space $X$, we let $P(X)$ denote the space of Borel probability measures equipped with the weak-* topology. Given a measure $\mu\in P(X)$, a barycenter (or Frechet mean) on $\mu$ is a minimizer of the average squared distance to points in the support of $\mu$; that is, an element of

[TABLE]

These metric barycenters are natural generalization of centers of mass of distributions on Euclidean space. As the function $y\mapsto\int_{X}d^{2}(x,y)d\mu(x)$ is in general highly non-convex, except under strong additional criteria (for example, that $X$ a simply connected, non-positively curved space), the minimizer may not be unique; that is, $b(\mu)$ may not be a singleton. Choosing a unique canonical minimizer in any reasonable sense is clearly impossible; for example, if $\mu$ is a uniform measure on the equator of the round sphere, the north and south pole are both barycenters and neither is any more natural than the other. To develop a canonical notion of barycenter, we utilize the geometry of $P(X)$ as follows.

Via the natural isometry $x\mapsto\delta_{x}$, one may isometrically embed the set $X$ into $P(X)$ equipped with Wasserstein metric

[TABLE]

where the infimum is over the set of all probability measures $\gamma$ on $X\times X$ with marginals $\mu$ and $\nu$. The minimization (1.2) is the well known Monge-Kantorovich problem, reviewed for example, in books of Villani [16, 17], and more recently Santambrogio [12]; our particular interest here lies in the fact that the Wasserstein distance extends the underlying geometry on $X$ to $P(X)$. It is also relevant here (and well known) that $P(X)$ with this metric inherits compactness from $X$, and that the Wasserstein metric topology coincides with the weak-* topology.

One can then consider instead the barycenter of the image measure $(x\mapsto\delta_{x})_{\#}\mu$ on the metric space $(P(X),W_{2})$, which amounts to finding a minimizer of

[TABLE]

The general study of barycenters in the Wasserstein space was initiated by Agueh-Carlier [1] when the underlying space $X=\mathbb{R}^{n}$ is Euclidean, and continued by the present authors to the setting where $X$ is a smooth Riemannian manifold [8]. These represent a natural, non-linear way to interpolate between several (or infinitely many) probability measures, and have received a great deal of attention in recent years, due in part to important applications in image processing and statistics; see, for example, the work of Rabin et al. [11] and Bigot and Klein [2] among others. Clearly, if $x\in BC(\mu)$ is a barycenter of $\mu$, then $\delta_{x}$ is a barycenter of $(x\mapsto\delta_{x})_{\#}\mu$, and so this minimization can be considered a relaxation of the barycenter problem on $X$. Although the later problem is on the infinite dimensional space $P(X)$ rather than $X$, it has the advantage of being a linear minimization. On the other hand, it certainly does not resolve the non-uniqueness issue; the minimizers of (1) are exactly those measures which are supported on $b(\mu)$, while only the Dirac masses supported on the same set correspond to classical barycenters.

On the other hand, one can try to define a canonical, distinguished minimizer of (1). For that purpose, throughout the paper we assume that $(X,d,{\bf{m}})$ satisfies a regularity condition (see Definition 2.1); that is, the minimizers of (1.2) in the definition of $W_{2}({\bf m},\nu)$ are realized as mappings pushing ${\bf m}$ forward to $\nu$. Examples of regular spaces $X$ include any smooth Riemannian manifold (with ${\bf m}$ absolutely continuous with respect to local coordinates) due to the Brenier-McCann theorem [3, 10], and more general (singular) metric measure spaces including the $CD(K,N)$ space in the sense of Lott-Villani-Sturm [9, 14, 15]; see the work of Cavaletti and Huesmann [4], Gigli [6], and Gigli, Rajala, and Sturm [7]. In this case, we show that one can pick the unique minimizer which is the most spread out, or the closest to the reference measure ${\bf m}$. This is the approach adopted in this paper.

For $\epsilon>0$, consider the function on $P(X)$ defined by

[TABLE]

It is not hard to establish using regularity of $X$, that $F_{\epsilon}$ has a unique minimizer $\mu_{\epsilon}$. In our main theorem (Thereom 2.2) we show that as $\epsilon\rightarrow 0$, this minimizer converges weakly-* to a unique measure $\nu$ which minimizes $W_{2}^{2}({\bf m},\cdot)$ among all minimizers of (1). We call this distinguished minimizer the Wasserstein regularized barycenter of $\mu$, and denote it by $B(\mu)$. This Wasserstein regularized barycenter is a weak notion of the metric barycenter; moreover, we verify that $B(\mu)$ is the unique minimizer of $\nu\mapsto W_{2}({\bf m},\nu)$ among all $\nu\in P(X)$ with $\mathop{\rm supp}\nu\subset b(\mu)$.

The mapping $\mu\mapsto B(\mu)$ induces a subtle dynamical structure on $P(X)$, and we go on to establish some notable properties of it: the mapping reduces the variance (Corollary 3.3), has fixed points (including Dirac masses, but also others, see Remark 2.3 and Example 3.5), has periodic orbits of period 2 (Examples 3.4 and 3.5) but not greater (Corollary 3.6) and is monotone in convex order (Corollary 3.7).

In the next section, we prove our main theorem regarding $B(\mu)$, while Section 3 is reserved for the development of various properties of the mapping $B:P(X)\rightarrow P(X)$.

2 Wasserstein regularized Riemannian barycenter $B(\mu)$: existence, uniqueness and characterization.

In this section, we prove that the Wasserstein regularized barycenter $B(\mu)$ of $\mu\in P(X)$ is well defined; that is, the weak-* limit $B(\mu)$ of the $\mu_{\epsilon}$ as $\epsilon\rightarrow 0$ exists. Moreover, we characterize $B(\mu)$ as the probability measure supported on the set $b(\mu)$ of barycentric points of $\mu$ which is closest to the reference measure ${\bf m}$ in the Wasserstein distance.

Definition 2.1** (Regularity of $X$).**

We say that the metric measure space $(X,d,{\bf m})$ is regular if for every $\nu\in P(M)$, any minimizer $\gamma$ in the definition (1.2) of $W_{2}({\bf m},\nu)$ is concentrated on the graph of a function; that is, $\gamma=(Id,T)_{\#}\bf m$ for a map $T:X\rightarrow X$.

As mentioned in the introduction, many metric measure spaces are regular, including any compact smooth Riemannian manifold where the references measure $\bf m$ is absolutely continuous with respect to volume, and other more singular spaces such as $CD(K,N)$ spaces. The case where ${\bf m}$ is (normalized) Riemannian volume is our motivating example.

It is well known that regularity ensures that the function $\nu\mapsto\epsilon W_{2}^{2}(\nu,{\bf m})$ is strictly convex with respect to linear interpolation between measures in $P(X)$ (see, e.g. [12, Proposition 7.19]). Since the functional $\nu\mapsto\int_{X}\int_{X}d^{2}(y,x)d\mu(x)d\nu(y)$ is linear, we see that $F_{\epsilon}$ in (1.4) is strictly convex in $\nu$ (with respect to linear interpolation); therefore its minimizer, $\mu_{\epsilon}$ is unique. Then, from the weak-* compactness of probability measures as $\epsilon\to 0$, a limit point of the $\mu_{\epsilon}$ exists. The following result establishes uniqueness of this limit point (that is, the limit exists) and a characterization of it.

Theorem 2.2** (Characterization of $B(\mu)$).**

As $\epsilon\rightarrow 0$, $\mu_{\epsilon}$ converges weak- to a unique limit $B(\mu)$ and we have*

[TABLE]

where $b(\mu)$ is the set of barycentric points defined in (1.1). Moreover, $B(\mu)$ is the unique minimizer of the functional $\nu\mapsto W_{2}(\nu,{\bf m})$ among $\nu\in P(X)$ with $\mathop{\rm supp}\nu\subset b(\mu)$, i.e.

[TABLE]

Proof.

For notational simplicity let us denote

[TABLE]

Step 1:

A standard continuity-compactness argument yields the existence of a minimizer $\bar{\mu}$ in (2.1). Now note that the set of probability measures supported on $b(\mu)$ is convex (with respect to linear interpolation) and so as the functional in (2.1) is strictly convex by [12, Proposition 7.19], the minimizer $\bar{\mu}$ is unique.

Step 2:

Now, let $\mu_{0}$ be any limit point of the minimizers $\mu_{\epsilon}$ of $F_{\epsilon}$ as $\epsilon\rightarrow 0$. We will show that $\mathop{\rm supp}\mu_{0}\subset b(\mu)$. Suppose $z\in\mathop{\rm supp}\mu_{0}$, but $z\not\in b(\mu)$. As the set $b(\mu)$ is closed, there exists $r>0$ such that for the metric ball $B_{r}(z)$ of radius $r$, $B_{2r}(z)\cap b(\mu)=\emptyset$, and, moreover, $\mu_{0}(B_{r}(z))>0$. Since $B_{2r}(z)$ is disjoint from $b(\mu)$, we have $\eta>0$ such that

[TABLE]

and so

[TABLE]

Fix a point $\bar{x}\in b(\mu)$ and define $\tilde{\mu}=\delta_{\bar{x}}$. Then,

[TABLE]

As $F_{\epsilon}(\nu)$ is continuous in both $\nu$ (with respect to the weak-* topology) and $\epsilon$, and $\mu_{0}$ is a limit point of the $\mu_{\epsilon}$ converges, this is a contradiction for small $\epsilon$, establishing $\mathop{\rm supp}\mu_{0}\subset b(\mu)$.

Step 3:

We now show that any limit point $\mu_{0}$ of the $\mu_{\epsilon}$ must coincide with $\bar{\mu}$; this will show that the limit $B(\mu)$ is well defined and equal to $\bar{\mu}$.

To see this, suppose by contradiction that there is a limit point $\mu_{0}\neq\bar{\mu}$ of the $\mu_{\epsilon}$. Then by steps 1 and 2, there is a $\delta>0$ with

[TABLE]

Choose $\epsilon>0$ sufficiently small so that $W_{2}(\mu_{0},\mu_{\epsilon})\leq\frac{\delta}{4D}$, where $D$ is the diameter of $X$. We have that

[TABLE]

and so

[TABLE]

which contradicts that $\mu_{\epsilon}$ is a minimizer of $F_{\epsilon}$, and therefore establishes that $\mu_{\epsilon}$ converges to $B(\mu)=\bar{\mu}$ as $\epsilon\rightarrow 0$.

Step 4.

Finally, we verify that $b(\mu)\cap\mathop{\rm supp}({\bf m})\subset\mathop{\rm supp}\bar{\mu}$. Suppose not; then there exists a barycenter point $\bar{x}\in b(\mu)$ and $r>0$ such that the metric ball $B_{r}(\bar{x})$ satisfies ${\bf m}(B_{r}(\bar{x}))>0$ and $\bar{\mu}(B_{2r}(\bar{x}))=0$. Now, let $\gamma\in\Gamma({\bf m},\bar{\mu})$ be an optimal transport plan from ${\bf m}$ to $\bar{\mu}$. Define $\tilde{\mu}$ as

[TABLE]

where $\pi^{2}$ is the projection $X\times X\ni(x,y)\mapsto y\in X$ and $\gamma_{B_{r}(\bar{x})\times X}$ is the restriction of $\gamma$ to the set $B_{r}(\bar{x})\times X$. Then, as $\bar{\mu}$ is supported on $b(\mu)$, so is $\tilde{\mu}$. Define $\tilde{\gamma}$, a transport plan from ${\bf m}$ to $\tilde{\mu}$, as

[TABLE]

Note this plan $\tilde{\gamma}$ modifies $\gamma$ by transporting the mass on $B_{r}(\bar{x})$ to the Dirac at $\bar{x}$. From the assumption $\bar{\mu}(B_{2r}(\bar{x})=0$, we see that

[TABLE]

Therefore, from the obvious inequality

[TABLE]

we get

[TABLE]

contradicting the characterization of $\bar{\mu}$ as the minimizer of $\nu\mapsto W_{2}^{2}({\bf m},\nu)$ among probability measures supported on $b(\mu)$. ∎

Remark 2.3** (Instability).**

The mapping $\mu\mapsto B(\mu)$ is highly unstable. To see this, consider uniform measure $\mu$ on the round sphere $X$ with ${\bf m}={\rm vol}$. Symmetry considerations easily imply that $B(\mu)=\mu$. Now, set $\mu^{\epsilon}:=\epsilon p_{n}+(1-\epsilon)\mu$, where $p_{n}$ is the north pole. Then for any $\epsilon>0$, it is easy to see that $B(\mu^{\epsilon})=\delta_{p_{n}}$. As $\mu^{\epsilon}$ is close to $\mu$, but $B(\mu^{\epsilon})=p_{n}$ is far from $B(\mu)=\mu$ in any reasonable topology, we conclude that $B$ is not stable.

3 Further properties of $B(\mu)$.

We now investigate some properties of $B(\mu)$. First, as an immediate corollary of the characterization of $B(\mu)$ in Theorem 2.2, we have the following result:

Corollary 3.1** (Support determines $B(\mu)$).**

If $\mathop{\rm supp}B(\mu)=\mathop{\rm supp}B(\nu)$, then $B(\mu)=B(\nu)$.

Nest, note that we can interpret (2.1) as characterizing $B(\mu)$ as the projection of the reference measure ${\bf m}$ to the set of measures supported on $b(\mu)$, with respect to Wasserstein distance. In the Riemannian setting, the following result makes this characterization more explicit.

Corollary 3.2**.**

Assume $X$ is Riemannian and smooth. For almost all $x$ (with respect to ${\rm vol}$), there is a unique $y\in{\rm argmin}_{y\in b(\mu)}d^{2}(x,y)$. Denoting the unique minimizer $y=:T(x)$, and assuming ${\bf m}$ is absolutely continuous with respect to volume, this $T$ is the optimal transport mapping from ${\bf m}$ to $B(\mu)$; in particular,

[TABLE]

Proof.

Set $f(x)=\min_{y\in b(\mu)}d^{2}(x,y)$. By a now standard argument of McCann [10], $f$ is Lipschitz and hence differentiable almost everywhere (with respect to ${\rm vol}$) by Rademacher’s theorem. Another argument in [10] implies that $x\mapsto d^{2}(x,y)$ is differentiable whenever $f$ is, and, as $f(x)-d^{2}(x,y)\leq 0$ for all $y$, with equality for $y\in{\rm argmin}_{y\in b(\mu)}d^{2}(x,y)$, we have, for all $x$ at which $f$ is differentiable and all $y\in{\rm argmin}_{y\in b(\mu)}d^{2}(x,y)$

[TABLE]

Equivalently, $y=\exp_{x}(2\nabla f(x)):=T(x)$; that is, $y$ is uniquely determined by $x$. This holds wherever $f$ is differentiable, and therefore ${\bf m}$-a.e.

Now, for any other $\nu$ supported on $b(\mu)$ letting $T_{\nu}$ be the optimal map from ${\bf m}$ to $\nu$, we have $d(x,T_{(}x))\leq d^{2}(x,T_{\nu}(x))$ for almost all $x$ and so

[TABLE]

which establishes minimality of $T_{\#}{\bf m}$ and therefore that $T_{\#}{\bf m}=B(\mu)$, by Theorem 2.2, and that $T$ is the optimal map between ${\bf m}$ and $B(\mu)$. ∎

Next, we note that, although $B(\mu)$ may not be supported on a single point, it is at least no more spread out than $\mu$, in the sense that it has lower variance, ${\rm var}(\mu):=\min_{y\in X}\int_{X}d^{2}(x,y)d\mu(x)$.

Corollary 3.3** (Variance reduction).**

For $\mu\in P(X)$,

[TABLE]

Moreover, the equality holds if and only if $\mathop{\rm supp}\mu\subset\mathop{\rm supp}B(B(\mu))$.

Proof.

Observe that ${\rm var}(\mu)=\int_{X}\int_{X}d^{2}(x,y)d\mu(x)dB(\mu)(y)$ since from Theorem 2.2, $\mathop{\rm supp}B(\mu)=b(\mu)$ where $b(\mu)$ is the set of barycenter points of $\mu$. Now note that

[TABLE]

The equality holds if and only if $\mu$ is a minimizer of $\nu\mapsto\int_{X}\int_{X}d^{2}(x,y)dB(\mu)(x)d\nu(y)$. This is equivalent to $\mathop{\rm supp}\mu\subset\mathop{\rm supp}B(B(\mu))$. ∎

The equality case above is illustrated in the following simple examples:

Example 3.4**.**

Let $X$ be the $n$-dimensional Riemannian round sphere with ${\bf m}={\rm vol}$, and let $\mu=\frac{1}{2}\delta_{p_{s}}+\frac{1}{2}\delta_{p_{n}}$, where $p_{s}$ and $p_{n}$ are the south and north poles, respectively. Then, $B(\mu)$ is the uniform probability measure on the equator, and $B(B(\mu))=\mu$.

Example 3.5**.**

Let $X=S_{1}$ be the circle, $\bf m$ be the normalized arc-length and $\mu=\sum_{i=1}^{N}\frac{1}{N}\delta_{x_{i}}$, where $\{x_{1},...,x_{N}\}$ are evenly spaced points on $X$. The set $b(\mu)$ of minimal points of the function $y\mapsto\sum_{i=1}^{N}\frac{1}{N}d^{2}(y,x_{i})$ depends on the parity of $N$:

If $N$ is odd, the function is minimized at each $x_{i}$, and so by rotational symmetry the regularized barycenter is $B(\mu)=\sum_{i=1}^{N}\frac{1}{N}\delta_{x_{i}}=\mu$; that is, $\mu$ is a fixed point of $B$. 2. 2.

If $N$ is even, the minimizing points are exactly those points $y_{i}$, $i=1,2,...N$ which are halfway in between two neighbouring $x_{i}^{\prime}s$, and so $B(\mu)=\sum_{i=1}^{N}\frac{1}{N}\delta_{y_{i}}$. An identical argument then yields $B(B(\mu))=\mu$.

As we see from the past two examples, it is possible that the operation $\mu\mapsto B(\mu)$ in $P(X)$ may have a periodic orbit. We next prove that no orbit can be periodic with period greater than two.

Corollary 3.6** (Period is at most $2$.).**

Suppose that $B^{N}(\mu)=\mu$ for some positive integer $N$. Then $B^{2}(\mu)=\mu$.

Proof.

The general inequality ${\rm var}(B(\mu))\leq{\rm var}(\mu)$ combined with periodicity easily implies that ${\rm var}(B^{k}(\mu))={\rm var}(\mu)$ for all positive integers $k$. In particular, ${\rm var}(\mu)={\rm var}(B(\mu))$, and Corollary 3.3 implies that $\mathop{\rm supp}(\mu)\subseteq\mathop{\rm supp}(B^{2}(\mu))\subseteq\mathop{\rm supp}(B^{4}(\mu))\subseteq...\subseteq\mathop{\rm supp}(B^{2N}(\mu))=\mathop{\rm supp}(\mu)$. As the first and last terms in this string of inclusions coincide, we must have equality throughout; in particular, $\mathop{\rm supp}(\mu)=\mathop{\rm supp}(B^{2}(\mu))$. Therefore, as $\mu=B^{N}(\mu)=B(B^{N-1}(\mu))$, we have

[TABLE]

and so Corollary 3.1 implies that $B(\mu)=B^{N-1}(\mu)$. Applying $B$ to both sides we arrive at

[TABLE]

This completes the proof. ∎

Finally, we record a version of Jensen’s inequality for the Wasserstein regularized barycenter, which might alternatively be interpreted as expressing a convex order between $B(\mu)$ and $\mu$ (see Remark 3.8 below). Recall that a function $\phi:X\to\mathbb{R}$ is said to be geodesically convex if for each geodesic segment $\sigma:[0,1]\to X$, the function $\phi(\sigma(t))$ is convex.

Corollary 3.7** (Monotonicity in convex order).**

Assume that $X$ is a Riemannian manifold and $\mu$ is absolutely continuous with respect to ${\rm vol}$. For any geodesically convex function $\phi$ on $X$, we have

[TABLE]

Proof.

First, we recall that the classical Jensen’s inequality extends to Riemannian manifolds (see, for instance, [5, Proposition 2]), asserting that, for any $y\in b(\mu)$, and geodesically convex function $\phi$,

[TABLE]

Now, as $\mathop{\rm supp}(B(\mu))\subseteq b(\mu)$, integrating (3.1) yields the desired result. ∎

We conclude the paper with a brief remark, offering some perspective on the preceding corollary.

Remark 3.8** (Martingales and convex order on Riemannian manifolds).**

Recall that a coupling $\pi$ between two probability measures $\mu$ and $\nu$ on $\mathbb{R}^{n}$ is a (discrete) martingale for $(\mu,\nu)$ if for $\mu$ almost every $x$,

[TABLE]

where $\pi_{x}$ represents the disintegration of $d\pi(x,y)=d\pi_{x}(y)d\mu(x)$ with respect to $\mu$. Strassen’s coupling theorem [13] asserts that there exists a martingale coupling of $\mu$ and $\nu$ if and only if $\nu$ dominates $\mu$ in convex order; that is $\int_{\mathbb{R}^{n}}\phi(x)d\mu(x)\leq\int_{\mathbb{R}^{n}}\phi(x)d\nu(x)$ for all convex $\phi:\mathbb{R}^{n}\rightarrow\mathbb{R}$.

On a metric space $X$, it is natural to define a martingale coupling of $\mu$, $\nu\in P(X)$ to be a coupling $d\pi(x,y)=d\pi_{x}(y)d\mu(x)$ such for $\mu$-a.e. $x$, we have

[TABLE]

By analog with the Euclidean case, we will say that $\mu$ dominates $\nu$ in (geodesically) convex order if we have $\int_{X}\phi(x)d\mu(x)\leq\int_{X}\phi(x)d\nu(x)$ for all geodesically convex $\phi$.

On a smooth Riemannian manifold, it is not hard to see (using (3.1)) that, if there exists a martingale coupling of $\nu$ and $\mu$ then $\mu$ dominates $\nu$ in convex order; that is, one implication of Strassen’s theorem extends to manifolds. The converse fails in general; on the sphere, for example, the only geodesically convex functions are constants, so any $\mu$ dominates any $\nu$ in convex order.

The preceding proposition asserts that on any manifold, $\mu$ dominates $B(\mu)$ in convex order. In this case, it is worth noting that product measure is a martingale between them. So the collection of pairs $\{(B(\mu),\mu)\}_{\mu\in P(X)}$ is a collection of marginals for which the conclusion of Strassen’s theorem extends to manifolds. We also note that when $B^{2}(\mu)=\mu$, product measure is a martingale with respect to either order; that is, it is a martingale for $(B(\mu),\mu)$ and for $(\mu,B(\mu))=(B(B(\mu)),B(\mu))$.