PRIMAL-GMM: PaRametrIc MAnifold Learning of Gaussian Mixture Models


We propose a ParametRIc MAnifold Learning (PRIMAL) algorithm for Gaussian mixtures models (GMM), operating under the assumption 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. In other words, we assume that GMMs are located on or near this manifold. The principal component analysis (PCA) served as the basis for modeling the generative processes for the priors, means, and covariance matrices. Each matrix's respective latent space and parametric mapping were used to represent these processes. The dependencies that exist between the latent spaces are then captured by a hierarchical latent space using either a linear or kernelized mapping. Minimizing the reconstruction error between ground-truth GMMs and manifold-generated GMMs, as measured by Kullback-Leibler Divergence, is how the function parameters and hierarchical latent space are learned (KLD). The insolvable KLD problem between GMMs is solved with the help of variational approximation, and a variational EM algorithm is developed in order to maximize the value of the objective function. Experiments conducted with synthetic data, flow cytometry analysis, eye-fixation analysis, and topic models demonstrate that PRIMAL learns a continuous and interpretable manifold of GMM distributions while simultaneously achieving a minimum reconstruction error.

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