Wasserstein barycenters and applications in image processing

In optimal transport, a Wasserstein barycenter is a probability measure that acts as a center of mass between two or more probability measures. It generalizes the notions of physical barycenters and geometric centroids.

Introduction

Motivation

Barycenters in physics and geometry are points that represent a notion of a mean of a number of objects. In astronomy, the barycenter is the center of mass of two or more objects that orbit each other, and in geometry, a centroid is the arithmetic mean of all the points in an object. Given countably many points $\{x_{i}\}_{i\in I}$ in $\mathbb {R} ^{n}$ with nonnegative weights $\{\lambda _{i}\}_{i\in I}$, the weighted $L^{2}$ barycenter of the points is the unique point $y$ minimizing $\sum _{i\in I}\lambda _{i}||y-x_{i}||_{2}^{2}$. Wasserstein barycenters attempt to capture this concept for probability measures by replacing the Euclidean distance with the Wasserstein distance of two probability measures, $W_{2}(\rho ,\mu )$.

Definition

Let $\Omega$ be a domain and ${\mathcal {P}}(\Omega )$ be the set of probability measures on $\Omega$. Given a collection of probability measures $\{\mu _{i}\}_{i\in I}$ and nonnegative weights $\{\lambda _{i}\}_{i\in I}$, we define a weighted barycenter of $\{\mu \}$ as any probability measure $\mu$ that minimizes  $\sum _{i\in I}\lambda _{i}W_{2}(\mu _{i},\mu )^{2}$ over the space $\mu \in {\mathcal {P}}(\Omega )$. Here $W_{2}$ denotes the 2-Wasserstein distance, which may be replaced with the $p$-Wasserstein distance, $W_{p}$, though this is not always as convenient.

The minimization problem above was originally introduced by Agueh and Carlier , who also proposed an alternative formulation of the problem above when there are finitely many measures $\{\mu _{i}\}_{i=1}^{N}$. Instead of considering the sum over all probability measures $\mu \in {\mathcal {P}}(\Omega )$, one can equivalently consider the optimization problem over all multi-marginal transport plans $\gamma \in {\mathcal {P}}(\Omega ^{N+1})$ whose push forwards satisfy Failed to parse (syntax error): {\displaystyle (\pi_i)_{#} \gamma = \mu_i} for $i\in \{1,\ldots ,N\}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\pi_0)_{#} \gamma = \mu} for some variable probability measure $\mu$.

Generalizations

Wasserstein barycenters are examples of Karcher and Fréchet means where the distance function used is the Wasserstein distance.