Wasserstein barycenters and applications in image processing: Difference between revisions
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===Definition=== | ===Definition=== | ||
Given points $\{x_i\}_{i \in I}$ in <math>R^n<\math> with nonnegative weights <math>\{\lambda_i \}_{i \in I}<\math>, the weighted <math>L^2<\math> barycenter of the points is the unique point <math>\{y\}<\math> minimizing <math>\sum_{i \in I} \lambda _i ||y - x_i||^2<\math>. To generalize this to Wasserstein spaces, let <math>\Omega<\math> be a domain and <math>P(\Omega)<\math> be the set of probability measures on <math>\Omega<\math>. Given a collection of probability measures <math>\{\mu_i/\rho_i\}_{i \in I}<\math> and nonnegative weights <math>\{\lambda_i\}_{i \in I}<\math>, we define a weighted barycenter of <math>\{\mu\}<\math> as any probability measure <math>\mu/\rho<\math> that minimizes | Given points $\{x_i\}_{i \in I}$ in <math> R^n <\math> with nonnegative weights <math>\{\lambda_i \}_{i \in I}<\math>, the weighted <math>L^2<\math> barycenter of the points is the unique point <math>\{y\}<\math> minimizing <math>\sum_{i \in I} \lambda _i ||y - x_i||^2<\math>. To generalize this to Wasserstein spaces, let <math>\Omega<\math> be a domain and <math>P(\Omega)<\math> be the set of probability measures on <math>\Omega<\math>. Given a collection of probability measures <math>\{\mu_i/\rho_i\}_{i \in I}<\math> and nonnegative weights <math>\{\lambda_i\}_{i \in I}<\math>, we define a weighted barycenter of <math>\{\mu\}<\math> as any probability measure <math>\mu/\rho<\math> that minimizes | ||
<math>\sum_{i \in I} \lambda_i W_2^2( \rho_i, \rho)^2<\math> | <math>\sum_{i \in I} \lambda_i W_2^2( \rho_i, \rho)^2<\math> | ||
over the space <math>\rho \in P(\Omega)<\math>. | over the space <math>\rho \in P(\Omega)<\math>. | ||
Revision as of 20:24, 11 February 2022
In optimal transport, a Wasserstein barycenter (Insert reference of Sant) is a probability measure 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
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 position of all the points in an object. Wasserstein barycenters attempt to
Definition
Given points $\{x_i\}_{i \in I}$ in <math> R^n <\math> with nonnegative weights <math>\{\lambda_i \}_{i \in I}<\math>, the weighted <math>L^2<\math> barycenter of the points is the unique point <math>\{y\}<\math> minimizing <math>\sum_{i \in I} \lambda _i ||y - x_i||^2<\math>. To generalize this to Wasserstein spaces, let <math>\Omega<\math> be a domain and <math>P(\Omega)<\math> be the set of probability measures on <math>\Omega<\math>. Given a collection of probability measures <math>\{\mu_i/\rho_i\}_{i \in I}<\math> and nonnegative weights <math>\{\lambda_i\}_{i \in I}<\math>, we define a weighted barycenter of <math>\{\mu\}<\math> as any probability measure <math>\mu/\rho<\math> that minimizes <math>\sum_{i \in I} \lambda_i W_2^2( \rho_i, \rho)^2<\math> over the space <math>\rho \in P(\Omega)<\math>.