Spectral Clustering, Bayesian Spanning Forest, and Forest Process.

Area of Science:

• Computational Statistics
• Machine Learning
• Data Mining

Background:

• Spectral clustering partitions data by minimizing graph-cut loss, avoiding explicit within-cluster distribution modeling.
• While effective, spectral clustering lacks direct methods for quantifying clustering uncertainty or extending models for complex applications.

Purpose of the Study:

• To develop a generative graphical model that bridges the gap in uncertainty quantification and model extensibility for spectral clustering.
• To introduce the Bayesian forest model as a novel approach for enhanced spectral clustering.

Main Methods:

• Proposed a Bayesian forest model, a generative graphical model leveraging the connection between forest posterior matrices and spectral clustering eigenvectors.
• Developed a 'forest process' as a graph-based extension to the urn process to induce distributions for the forest.
• Derived a Markov chain Monte Carlo (MCMC) algorithm for posterior estimation.

Main Results:

• The posterior connecting matrix in the forest model shares leading eigenvectors with normalized spectral clustering.
• Demonstrated superior performance of the proposed Bayesian forest model compared to existing spectral clustering algorithms.
• Illustrated model-based extensions applicable to high-dimensional and multi-view clustering, particularly for image data.

Conclusions:

• The Bayesian forest model effectively addresses limitations of traditional spectral clustering by providing uncertainty quantification and model flexibility.
• The proposed method offers a robust and extensible framework for advanced data clustering tasks, including complex image analysis.