Fréchet mean

Summary

The Fréchet mean generalises the average to data in a metric space: it is a point that minimises the expected squared distance to the data. In Euclidean space it is the usual mean. It also makes sense for points on a sphere, on a Riemannian manifold, or in spaces of trees, shapes or covariance matrices. In these spaces adding and dividing points is not defined. It need not exist or be unique, and the conditions for uniqueness involve the curvature of the space. It is named after Maurice Fréchet; local minimisers on Riemannian manifolds are also called Karcher means.