Multiple Manifolds Learning Framework Based on Hierarchical Mixture Density Model

Several manifold learning techniques have been developed to learn, given a data, a single lower dimensional manifold providing a compact representation of the original data. However, for complex data sets containing multiple manifolds of possibly of different dimensionalities, it is unlikely that the existing manifold learning approaches can discover all the interesting lower-dimensional structure. We therefore introduce a hierarchical manifolds learning framework to discover a variety of the underlying low dimensional structures. The framework is based on hierarchical mixture latent variable model, in which each submodel is a latent variable model capturing a single manifold. We propose a novel multiple manifold approximation strategy used for initialization of our hierarchical model. The technique is first verified on artificial data with mixed 1-, 2and 3-dimensional structures. It is then used to automatically detect lower-dimensional structures in disrupted satellite galaxies.