PRIMAL-GMM : PaRametrIc MAnifold Learning of Gaussian Mixture Models

We propose a ParametRIc MAnifold Learning (PRIMAL) algorithm for Gaussian Mixtures Models (GMM), assuming that GMMs lie on or near to a manifold of probability distributions that is generated from a low-dimensional hierarchical latent space through parametric mappings. Inspired by Principal Component Analysis (PCA), the generative processes for priors, means and covariance matrices are modeled by their respective latent space and parametric mapping. Then, the dependencies between latent spaces are captured by a hierarchical latent space by a linear or kernelized mapping. The function parameters and hierarchical latent space are learned by minimizing the reconstruction error between ground-truth GMMs and manifold-generated GMMs, measured by Kullback-Leibler Divergence (KLD). Variational approximation is employed to handle the intractable KLD between GMMs and a variational EM algorithm is derived to optimize the objective function. Experiments on synthetic data, flow cytometry analysis, eye-fixation analysis and topic models show that PRIMAL learns a continuous and interpretable manifold of GMM distributions and achieves a minimum reconstruction error.