PRIMAL-GMM: PaRametrIc MAnifold Learning of Gaussian Mixture Models

Area of Science:

• Machine Learning
• Statistical Modeling
• Probability Theory

Background:

• Gaussian Mixture Models (GMMs) are widely used for data clustering and density estimation.
• Existing methods often struggle to capture the underlying structure of GMMs in a low-dimensional space.
• The manifold hypothesis suggests that complex data distributions can often be represented on lower-dimensional manifolds.

Purpose of the Study:

• To develop a novel algorithm, Parametric Manifold Learning (PRIMAL), for learning manifolds of GMMs.
• To model GMMs as arising from a low-dimensional hierarchical latent space via parametric mappings.
• To enable the discovery of continuous and interpretable GMM manifolds.

Main Methods:

• PRIMAL models generative processes for GMM parameters (priors, means, covariances) using latent spaces and parametric mappings.
• A hierarchical latent space captures dependencies, employing linear or kernelized mappings.
• Kullback-Leibler Divergence (KLD) measures reconstruction error, optimized via variational approximation and a variational EM algorithm.

Main Results:

• PRIMAL successfully learns a continuous and interpretable manifold of GMM distributions.
• The algorithm achieves a minimum reconstruction error on synthetic and real-world datasets.
• Demonstrated effectiveness across diverse applications including flow cytometry, eye-fixation analysis, and topic models.

Conclusions:

• PRIMAL provides an effective framework for manifold learning in the context of GMMs.
• The learned manifold offers insights into the structure and relationships between GMMs.
• This approach enhances the representational power and interpretability of GMMs in various data analysis tasks.