Spectral Clustering: Algorithm Implementation and Comparative Analysis

Spectral clustering is a clustering algorithm capable of identifying arbitrarily shaped sample spaces while converging to global optimal solutions. The core methodology involves performing eigen decomposition on sample data similarity matrices to obtain eigenvectors for clustering assignments. This implementation compares several clustering algorithms, including Q-matrix clustering (utilizing modularity optimization), k-means clustering (with Lloyd's algorithm), first eigencomponent clustering (based on dominant eigenvectors), second generalized eigencomponent clustering (using generalized eigenvalue problems), shared data generation (creating synthetic datasets via Gaussian mixtures), and neighborhood matrix generation (constructing k-nearest neighbor graphs). The comparative analysis enables better understanding of spectral clustering's performance characteristics and applicability, facilitating optimal algorithm selection for specific data processing requirements. Key implementation aspects include Laplacian matrix normalization techniques and eigenvalue thresholding for cluster determination.