Wasserstein barycenters and applications in image processing: Difference between revisions
No edit summary |
No edit summary |
||
Line 5: | Line 5: | ||
==Introduction== | ==Introduction== | ||
===Motivation=== | ===Motivation=== | ||
Barycenters in physics and geometry are points that represent a notion of a [https://en.wikipedia.org/wiki/Mean 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 [https://en.wikipedia.org/wiki/Arithmetic_mean arithmetic mean] of all the points in an object. Given countably many points <math>\{x_i\}_{i \in I}</math> 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||_2^2</math>. Wasserstein barycenters attempt to capture this concept for probability measures by replacing the [https://en.wikipedia.org/wiki/Euclidean_distance Euclidean distance] with the [https://en.wikipedia.org/wiki/Wasserstein_metric Wasserstein distance] of two probability measures, <math>W_2(\rho, \mu)</math>. | Barycenters in physics and geometry are points that represent a notion of a [https://en.wikipedia.org/wiki/Mean 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 [https://en.wikipedia.org/wiki/Arithmetic_mean arithmetic mean] of all the points in an object. Given countably many points <math>\{x_i\}_{i \in I}</math> in <math>\mathbb{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||_2^2</math>. Wasserstein barycenters attempt to capture this concept for probability measures by replacing the [https://en.wikipedia.org/wiki/Euclidean_distance Euclidean distance] with the [https://en.wikipedia.org/wiki/Wasserstein_metric Wasserstein distance] of two probability measures, <math>W_2(\rho, \mu)</math>. | ||
===Definition=== | ===Definition=== |
Revision as of 21:09, 11 February 2022
In optimal transport, a Wasserstein barycenter (Insert reference of Sant) 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 in with nonnegative weights , the weighted barycenter of the points is the unique point minimizing . Wasserstein barycenters attempt to capture this concept for probability measures by replacing the Euclidean distance with the Wasserstein distance of two probability measures, .
Definition
Let be a domain and be the set of probability measures on . Given a collection of probability measures and nonnegative weights , we define a weighted barycenter of as any probability measure that minimizes over the space . Here denotes the 2-Wasserstein distance.
Existence and Uniqueness
Other spaces
Applications
Barycenters in Image processing
Generalizations
Wasserstein barycenters are examples of Karcher and Fr%C3%A9chet means where the distance function used is the Wasserstein distance.