Poincaré Fréchet mean

Abstract

Generalizing the Fréchet mean from the Euclidean metric is not able to properly capture the geometric characteristics of many non-trivial operations, such as the non-dot inner product and non-Euclidean gradients defined on the manifold. One effective solution is to derive its hyperbolic representations in the Poincaré or hyperboloid model. Our goodness-of-fit testing shows that the Poincaré Fréchet mean achieves much lowerχ2power than that of the hyperboloid and typical non-linear kernels with regard to parameter perturbations. However, recent advanced optimization solvers, such as Riemannian gradient descent and minimizing upper bound, may result in imprecise convergences. This paper presents an (1−ϵ)-approximation approach to search a core-set on the Poincaré model, reducing deviations of the Poincaré Fréchet mean to its optimum. A hierarchical splitting algorithm that implicitly explores the hyperbolic representations for an arbitrary manifold is then presented. Experiments show that the (1−ϵ) Poincaré Fréchet mean adopted in hierarchical splitting, achieves better representations than Euclidean, kernel, and Lorentzian Fréchet means in graph and image data.