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A regularized version of Mixture Models is proposed to learn a principal graph from a distribution of <inline-formula><tex-math notation="LaTeX">$D$</tex-math><alternatives><mml:math><mml:mi>D</mml:mi></mml:math><inline-graphic xlink:href="bonnaire-ieq1-3124973.gif"/></alternatives></inline-formula>-dimensional datapoints. In the particular case of manifold learning for ridge detection, we assume that the underlying structure can be modeled as a graph acting like a topological prior for the Gaussian clusters turning the problem into a maximum <italic>a posteriori</italic> 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.

Regularization of Mixture Models for Robust Principal Graph Learning