Regularization of Mixture Models for Robust Principal Graph Learning

TL;DR

This paper introduces a regularized mixture model approach for learning principal graphs from high-dimensional data, incorporating robustness to outliers and heteroscedasticity, with efficient EM-based parameter estimation.

Contribution

It presents a novel regularization framework for mixture models that enables robust principal graph learning with guaranteed convergence and cycle incorporation.

Findings

Efficient EM algorithm with polynomial convergence guarantees.

Robustness to outliers and heteroscedasticity in manifold sampling.

Extension of minimum spanning tree prior with random sub-sampling for cycle detection.

Abstract

A regularized version of Mixture Models is proposed to learn a principal graph from a distribution of $D$-dimensional data points. In the particular case of manifold learning for ridge detection, we assume that the underlying manifold can be modeled as a graph structure acting like a topological prior for the Gaussian clusters turning the problem into a maximum a posteriori estimation. Parameters of the model are iteratively estimated through an Expectation-Maximization procedure making the learning of the structure computationally efficient with guaranteed convergence for any graph prior in a polynomial time. We also embed in the formalism a natural way to make the algorithm robust to outliers of the pattern and heteroscedasticity of the manifold sampling coherently with the graph structure. The method uses a graph prior given by the minimum spanning tree that we extend using random sub-samplings of the dataset to take into account cycles that can be observed in the spatial distribution.