Geodesic Finite Mixture Models

The use of Riemannian manifolds and their statistics has recently gained popularity in a wide range of applications involving non-linear data modeling. For instance, they have been used to model shape changes in the brain [1] and human motion [3]. In this work we tackle the problem of approximating the Probability Density Function (PDF) of a potentially large dataset that lies on a known Riemannian manifold. We address this by creating a completely data-driven algorithm consistent with the manifold, i.e., an algorithm that yields a PDF defined exclusively on the manifold. In the proposed finite mixture model, we simultaneously consider multiple tangent spaces, distributed along the whole manifold as seen in Fig. 1. We draw inspiration on the unsupervised Expectation Maximization (EM) algorithm from [2], which given data lying in an Euclidean space, automatically computes the number of model components that Minimize a Message Length (MML) cost. By representing each component as a distribution on the tangent space at its corresponding mean on the manifold, we are then able to generalize the algorithm to Riemannian manifolds and at the same time mitigate the accuracy loss produced when using a single tangent space. Given an input dataset, [2] starts by randomly initializing a large number of components. During the Maximization (M) step, the MML criterion is used to annihilate those components not well supported by the data. In addition, upon EM convergence, the least probable mixture component is also forcibly annihilated and the algorithm continues until a minimum number of components is reached. In order to extend [2] to Riemannian manifolds, we define each mixture component as a normal distribution on its own tangent space TμkM, with a mean μk and a concentration matrix Γk = Σ −1 k :