Geodesic Finite Mixture Models

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There are many cases in which data is found to be distributed on a Riemannian manifold. In these cases, Euclidean metrics are not applicable and one needs to resort to geodesic distances consistent with the manifold geometry. For this purpose, we draw inspiration on a variant of the expectation-maximization algorithm, that uses a minimum message length criterion to automatically estimate the optimal number of components from multivariate data lying on an Euclidean space. In order to use this approach on Riemannian manifolds, we propose a formulation in which each component is defined on a different tangent space, thus avoiding the problems associated with the loss of accuracy produced when linearizing the manifold with a single tangent space. Our approach can be applied to any type of manifold for which it is possible to estimate its tangent space.

Overview of the Geodesic Finite Mixture Model By using multiple tangent spaces, one for each mixture, as shown on the left, we are able to minimize the approximation error of the linearization. As shown in the paper, using multiple tangent spaces allows much better accuracy in the estimation of the Probability Density function of the data in comparison to using a single tangent space or other manifold-specific distributions such as the von Mises-Fisher distributions for the sphere manifold.

We have released the code necessary for anyone to be able to reproduce the results in the paper. We have hopes that they will be useful and others will be able to find other manifolds in which the GFMM can be applied for interesting results.

Below we show the supplemental material provided with the paper. It is roughly a 2 minute video showing the expectation-maximization optimization process on two different synthetic scenarios presented in the paper.

Feel free to contact me if you have any questions or use the algorithm.



  • Lie Algebra-Based Kinematic Prior for 3D Human Pose Tracking
    • Lie Algebra-Based Kinematic Prior for 3D Human Pose Tracking
    • Edgar Simo-Serra, Carme Torras, Francesc Moreno-Noguer
    • International Conference on Machine Vision Applications (MVA) [best paper], 2015


  • Geodesic Finite Mixture Models
    • Geodesic Finite Mixture Models
    • Edgar Simo-Serra, Carme Torras, Francesc Moreno-Noguer
    • British Machine Vision Conference (BMVC), 2014

Source Code

  • GFMM