Wasserstein Barycenter Overview

• Wasserstein barycenter is a probability measure that minimizes the average squared Wasserstein distance to a collection of measures, generalizing classical means.
• It extends existence and uniqueness results to non-compact, non-smooth, and infinite-dimensional spaces using convex analysis and gradient flow approaches.
• Synthetic curvature-dimension conditions and Jensen-type inequalities underpin new functional and geometric inequalities in optimal transport.

A Wasserstein barycenter is a probability measure that minimizes the average optimal transport (Wasserstein) distance squared to a prescribed collection of input measures. The barycenter provides a canonical way to interpolate or average measures using the geometry induced by the Wasserstein metric, and encodes a broad generalization of classical means to spaces of probability measures. In recent years, the paper of Wasserstein barycenters has expanded far beyond smooth, finite-dimensional, and compact settings, entering domains of non-compact, non-smooth, and infinite-dimensional metric measure spaces. Advances include foundational results on existence, uniqueness, and absolute continuity in highly general settings, synthetic curvature-dimension conditions adapted to barycentric convexity, and deep connections to gradient flows, geometric inequalities such as Jensen’s and Brunn-Minkowski, and the structural analysis of optimal transport.

1. Generalized Existence, Uniqueness, and Absolute Continuity

Given a metric space $(X, d)$ (possibly extended and non-compact) and a probability measure $\Omega$ on $(\mathcal{P}(X), W_2)$ with finite variance, a measure $\bar{\mu}$ is a Wasserstein barycenter if

$$\int_{\mathcal{P}(X)} W_2^2(\mu, \bar{\mu})\, d\Omega(\mu) = \min_{\nu \in \mathcal{P}(X)} \int_{\mathcal{P}(X)} W_2^2(\mu, \nu)\, d\Omega(\mu),$$

where $W_2$ is the $L^2$-Kantorovich distance. Existence is guaranteed under very mild assumptions, including for metric measure spaces with synthetic lower Ricci curvature bounds and upper dimension bounds (RCD(K,N) spaces), abstract Wiener spaces, and certain configuration spaces over Riemannian manifolds. Uniqueness of the barycenter holds if the reference space is RCD(K,0) and the barycenter is absolutely continuous. If the entropy functional is integrable, absolute continuity of the barycenter with respect to an appropriate reference measure is preserved, extending the theory beyond Euclidean and Riemannian contexts.

2. Extension to Non-compact, Non-smooth, and Infinite-dimensional Spaces

A distinguishing feature of recent advances is the extension of Wasserstein barycenter theory beyond locally compact or smooth spaces. The existence and regularity theory is robust even for infinite-dimensional settings such as abstract Wiener spaces. Key technical tools enabling this generalization are convex analysis and the evolution variational inequality (EVI) framework for gradient flows in metric measure spaces. Notably, the EVI framework enables a dimension-free, metric-based calculus independent of local compactness or smooth structure, making the barycenter well-defined and uniquely characterized even in extended or singular geometric contexts.

3. Synthetic Convexity and Jensen's Inequality for Barycenters

A central analytic feature is a synthetic Jensen’s inequality for Wasserstein barycenters, which provides an abstract convexity bridge: $$F(\bar{x}) \leq \int F(x)\, d\mu(x) - \frac{K}{2} \operatorname{Var}(\mu),$$ for barycentric-convex functionals $F$, where $\bar{x}$ is a barycenter and $\operatorname{Var}(\mu)$ denotes the minimum mean squared Wasserstein variance. For the entropy functional $\mathrm{Ent}_m$ relative to a reference measure $m$, the Wasserstein Jensen’s inequality (WJI) takes the form: $$\mathrm{Ent}_m(\bar{\mu}) \leq \int_{\mathcal{P}(X)} \mathrm{Ent}_m(\mu)\, d\Omega(\mu) - \frac{K}{2}\int_{\mathcal{P}(X)} W_2^2(\mu, \bar{\mu})\, d\Omega(\mu).$$ A dimensional Jensen's inequality is also established, involving curvature-dimension distortion coefficients $\sigma_{K,N}$, $\tau_{K,N}$, and an exponential entropy weight $U_N(\mu) = e^{- \mathrm{Ent}_m(\mu)/N}$: $$\int W_2^2(\mu, \bar{\mu})\, \sigma_{K,N}(W_2(\mu, \bar{\mu}))\, U_N(\mu)\, d\Omega(\mu) \leq U_N(\bar{\mu}) \int W_2^2(\mu, \bar{\mu})\, \tau_{K,N}(W_2(\mu, \bar{\mu}))\, d\Omega(\mu).$$

4. Synthetic Barycenter-Curvature-Dimension (BCD) Condition

A key conceptual development is the introduction of the Barycenter-Curvature-Dimension (BCD) condition, a synthetic curvature-dimension condition tailored to barycenters:

• BCD(K,∞):

For any finitely supported $\Omega$ on $\mathcal{P}(X)$, there exists a barycenter $\bar{\mu}$ such that

$$\mathrm{Ent}_m(\bar{\mu}) \leq \int \mathrm{Ent}_m(\mu)\, d\Omega(\mu) - \frac{K}{2} \operatorname{Var}(\Omega).$$

• BCD(K,N):

There exists a barycenter $\bar{\mu}$ such that

$$\int_{\mathcal{P}(X)} \frac{W_2^2(\mu, \bar{\mu})}{\tau_{K,N}(W_2(\mu, \bar{\mu}))} U_N(\mu) d\Omega(\mu) \leq U_N(\bar{\mu}) \int_{\mathcal{P}(X)} \frac{W_2^2(\mu, \bar{\mu})}{\sigma_{K,N}(W_2(\mu, \bar{\mu}))} d\Omega(\mu),$$

with the same exponential entropy weight.

The BCD condition is strictly weaker than the traditional Lott--Sturm--Villani CD(K,N) condition, requiring only barycentric convexity. Critically, the BCD(K,N) condition is stable under measured Gromov--Hausdorff convergence; thus, limits of spaces satisfying BCD(K,N) preserve the validity of barycenter theory and convexity inequalities.

5. Functional and Geometric Inequalities for Barycenters

Several new inequalities anchored in barycenter geometry are derived:

• Multi-marginal Brunn–Minkowski Inequality: For BCD(0,N) spaces, if $E$ is the set of barycenters associated to sets $E_i$ with weights $\lambda_i$,

$m(E) \geq \prod_{i=1}^n m(E_i)^{\lambda_i}.$

• Functional Blaschke–Santaló Inequality: On BCD(1,∞) spaces, for suitable functions $f_i$,

$$\left( \prod_{i=1}^k \int e^{f_i} dm \right) \leq \exp\left( \inf_{x_1, ..., x_k} \sum_{i=1}^k f_i(x_i) - \frac{1}{2k} \inf_x \sum_{i=1}^k d^2(x, x_i) \right).$$

Additional results guarantee the existence and uniqueness of Monge solutions for multi-marginal optimal transport with barycentric cost in general BCD-structured spaces, yielding barycenters that are also absolutely continuous even on non-compact or infinite-dimensional domains.

6. Implications for Optimal Transport and Metric Geometry

The suite of results outlined above expands the reach of Wasserstein barycenter theory to non-smooth, non-compact, infinite-dimensional, and singular metric measure spaces. By employing the BCD condition—a synthetic, flexible Ricci lower bound formulated via barycentric convexity—convexity and functional inequalities become accessible on spaces far from the classical Lott--Sturm--Villani paradigm. Gradient flow techniques and the EVI framework are central, allowing barycenter analysis to fully exploit the variational structure of the Wasserstein space, independently of smooth geometric prerequisites. These advances impact the analysis and convergence of metric measure spaces, the structure of multi-marginal optimal transport, and the development of convexity inequalities in non-classical settings.

7. Key Formulas and Generalization

The central mathematical objects and relations governing Wasserstein barycenters in the general setting are:

• Barycenter Definition:

$$\bar{\mu} \in \operatorname*{arg\,min}_{\nu \in \mathcal{P}(X)} \int_{\mathcal{P}(X)} W_2^2(\mu, \nu)\, d\Omega(\mu)$$

• Variance:

$$\operatorname{Var}(\Omega) = \inf_{\nu \in \mathcal{P}(X)} \int_{\mathcal{P}(X)} W_2^2(\mu, \nu)\, d\Omega(\mu)$$

• Jensen-Type Inequalities:

$$F(\bar{x}) \leq \int F(x)\, d\mu(x) - \frac{K}{2} \operatorname{Var}(\mu)$$

• Wasserstein Jensen (WJI) for Entropy:

$$\mathrm{Ent}_m(\bar{\mu}) \leq \int \mathrm{Ent}_m(\mu)\, d\Omega(\mu) - \frac{K}{2} \int W_2^2(\mu, \bar{\mu})\, d\Omega(\mu)$$

• Dimensional Jensen's Inequality:

$$\int W_2^2(\mu, \bar{\mu})\, \sigma_{K,N}\, U_N(\mu)\, d\Omega(\mu) \leq U_N(\bar{\mu}) \int W_2^2(\mu, \bar{\mu})\, \tau_{K,N}\, d\Omega(\mu)$$

where $U_N(\mu) = e^{- \mathrm{Ent}_m(\mu)/N}$.

These formulas underpin the structure of barycenters and the analytic and geometric properties derived from synthetic curvature assumptions and extend convexity geometry to a vast array of new settings.

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In summary, Wasserstein barycenter theory has been expanded to a highly general setting by introducing a synthetic barycenter-curvature-dimension condition and establishing robust generalizations of existence, uniqueness, absolute continuity, and convexity inequalities. This framework encompasses RCD spaces, abstract infinite-dimensional constructs, and is stable under metric convergence, furnishing a powerful analytic and geometric toolkit for the paper of optimal transport and measure-valued interpolation in non-classical metric measure spaces (Han et al., 2 Dec 2024).