## Centroids and statistical centroids (and other centers: medians, circumcenters, etc.)

In general, let us define a centroid as the minimizer of the average distance of a center object to a given collection of objects. For example, the Euclidean centroid minimizes the average squared Euclidean distance to a point set (= minimize the set variance wrt to a center), and is well-known to be the center of mass, the arithmetic mean of the points. When the collection of objects is a set of distributions and the distance a statistical distance, we get a statistical centroid (a probability distribution). In statistics, distances can be metric ones (eg, the total variation or Wasserstein distances) or non-metric thrice differentiable ones (called divergences like the Kullback-Leibler divergence or the Bhattacharrya divergence, and more generally f-divergences). Computing centroids are essential for center-based clustering a la k-means (or variational k-means). Here is a collection of centroids that we have studied: