A Wasserstein barycenter is an average of distributions formed in transport geometry rather than by mixing their densities. It is the Fréchet mean of probability measures and links geodesics, multi-marginal transport, convexity, and statistical estimation.
The Fréchet functional
Definition. Given \(\mu_1,\ldots,\mu_N\in\mathcal P_2(\mathbb R^d)\) and weights \(\lambda_i>0\), \(\sum_i\lambda_i=1\), define \[ \mathcal V(\nu)=\frac12\sum_{i=1}^N\lambda_iW_2^2(\nu,\mu_i). \] Any minimizer \(\bar\mu\) of \(\mathcal V\) is a Wasserstein barycenter.
Existence and uniqueness. A barycenter always exists. If at least one \(\mu_i\) is absolutely continuous, then the barycenter is unique; under standard hypotheses it is also absolutely continuous.
Existence. Let \((\nu_n)\) be minimizing. For any fixed \(i\), \[ W_2(\nu_n,\delta_0)\le W_2(\nu_n,\mu_i)+W_2(\mu_i,\delta_0), \] so bounded objective gives a uniform second-moment bound. Tightness and moment control yield a \(W_2\)-convergent subsequence. Lower semicontinuity of each squared distance makes its limit a minimizer.
Uniqueness mechanism. Suppose \(\mu_j\) is absolutely continuous. Interpolate linearly between two candidate barycenters through a generalized geodesic with base \(\mu_j\), obtained by gluing their unique optimal maps from \(\mu_j\). The term \(W_2^2(\cdot,\mu_j)\) is strictly convex along this interpolation unless the endpoints coincide, while the remaining terms are convex. Two distinct minimizers would therefore produce a midpoint with strictly smaller objective.
First-order optimality
Barycenter equation. Suppose the barycenter \(\bar\mu\) is absolutely continuous and let \(T_i\) be the Brenier map from \(\bar\mu\) to \(\mu_i\). Then \[ \sum_{i=1}^N\lambda_iT_i(x)=x \qquad\text{for }\bar\mu\text{-almost every }x. \]
Proof. Perturb \(\bar\mu\) by \(S_\varepsilon=\operatorname{id}+\varepsilon\xi\), with smooth compactly supported \(\xi\). Coupling \(S_\varepsilon(x)\) with \(T_i(x)\) gives \[ W_2^2(S_{\varepsilon\#}\bar\mu,\mu_i) \le\int|x+\varepsilon\xi(x)-T_i(x)|^2\,\mathrm{d}\bar\mu(x). \] Differentiate the upper bound at \(\varepsilon=0\). Minimality for positive and negative \(\varepsilon\) gives \[ \int \xi(x)\cdot\sum_i\lambda_i(x-T_i(x))\,\mathrm{d}\bar\mu(x)=0 \] for every \(\xi\). Hence the vector field in parentheses vanishes almost surely.
Multi-marginal formulation
Multi-marginal representation. Let \[ \bar x_\lambda=\sum_i\lambda_ix_i. \] Then \[ \inf_{\nu}\sum_i\lambda_iW_2^2(\nu,\mu_i) = \inf_{\gamma\in\Pi(\mu_1,\ldots,\mu_N)} \int\sum_i\lambda_i|x_i-\bar x_\lambda|^2\,\mathrm{d}\gamma. \] If \(\gamma^\star\) solves the right side, then \((\bar x_\lambda)_\#\gamma^\star\) is a barycenter.
Proof. Given a candidate \(\nu\), glue optimal couplings of \(\nu\) with all \(\mu_i\) conditionally on the barycenter variable \(z\). For every \(x_1,\ldots,x_N,z\), \[ \sum_i\lambda_i|x_i-z|^2 = \sum_i\lambda_i|x_i-\bar x_\lambda|^2+|z-\bar x_\lambda|^2. \] The minimum over \(z\) is attained at \(\bar x_\lambda\), giving one inequality. Conversely, any multi-marginal coupling and \(z=\bar x_\lambda\) induce couplings of its law with each \(\mu_i\), giving the other inequality.
One-dimensional barycenters
Quantile formula. On \(\mathbb R\), the unique \(W_2\)-barycenter satisfies \[ F_{\bar\mu}^{-1}(u)=\sum_{i=1}^N\lambda_iF_{\mu_i}^{-1}(u) \quad\text{for almost every }u\in(0,1). \]
Proof. By the one-dimensional formula, \[ \mathcal V(\nu)=\frac12\int_0^1 \sum_i\lambda_i|F_\nu^{-1}(u)-F_{\mu_i}^{-1}(u)|^2\,\mathrm{d}u. \] Pointwise minimization in \(F_\nu^{-1}(u)\) gives the weighted average. It remains nondecreasing because it is a positive combination of nondecreasing functions, so it is a valid quantile.
Gaussian barycenters
Gaussian barycenter. The barycenter of nondegenerate Gaussian laws \(N(m_i,\Sigma_i)\) is \(N(\bar m,\bar\Sigma)\), where \[ \bar m=\sum_i\lambda_im_i \] and \(\bar\Sigma\) is the unique positive-definite solution of \[ \bar\Sigma = \sum_i\lambda_i (\bar\Sigma^{1/2}\Sigma_i\bar\Sigma^{1/2})^{1/2}. \]
The equation is the matrix form of \(\sum_i\lambda_iT_i(x)=x\). If the covariance matrices commute, it simplifies to \[ \bar\Sigma^{1/2}=\sum_i\lambda_i\Sigma_i^{1/2}. \]
Stability of barycenters
Stability. Suppose \(\mu_{i,n}\to\mu_i\) in \(W_2\) and let \(\bar\mu_n\) be corresponding barycenters with fixed weights. Then \((\bar\mu_n)\) is relatively compact in \(W_2\), and every limit point is a barycenter of \((\mu_i)\). If the limiting barycenter is unique, then \(\bar\mu_n\to\bar\mu\) in \(W_2\).
Proof. Compare \(\mathcal V_n(\bar\mu_n)\) with \(\mathcal V_n(\mu_{1,n})\) to obtain a uniform second-moment bound. Compactness follows. The triangle inequality implies locally uniform convergence of \(W_2^2(\nu,\mu_{i,n})\) on moment-bounded sets, so the liminf inequality and recovery at any fixed \(\nu\) identify each limit as a minimizer. Uniqueness removes subsequences.
Population Fréchet means
If \(\mathsf Q\) is a probability distribution on \(\mathcal P_2(\mathbb R^d)\), a population barycenter minimizes \[ \nu\longmapsto\frac12\int W_2^2(\nu,\mu)\,\mathrm{d}\mathsf Q(\mu). \] The empirical barycenter replaces \(\mathsf Q\) by sampled distributions. Existence, uniqueness, consistency, and limit distributions are therefore Fréchet-mean questions in a nonlinear metric space; page 21 returns to their statistical theory.