MATLAB Code Implementation of Spectral Clustering

Spectral clustering is a graph theory-based clustering method that treats data points as vertices in a graph. It constructs an adjacency matrix by calculating pairwise similarities between data points, performs eigendecomposition on the graph Laplacian matrix, and finally applies traditional clustering (such as K-means) in the low-dimensional feature space. MATLAB provides an ideal environment for implementing spectral clustering algorithms due to its comprehensive matrix operations and linear algebra toolbox. A complete spectral clustering toolbox typically requires the following key modules: Similarity Matrix Computation Module: This module calculates pairwise similarities between data points. Common methods include Gaussian kernel functions and cosine similarity. The implementation should support different similarity metrics and allow parameter customization (e.g., bandwidth selection for Gaussian kernels), typically using vectorized operations for efficiency. Graph Laplacian Matrix Construction Module: Constructs the graph Laplacian matrix from the similarity matrix. Users can choose between unnormalized Laplacian, symmetric normalized Laplacian, or random walk normalized Laplacian. The implementation should handle diagonal degree matrices and matrix normalization operations. Eigendecomposition Module: Performs eigendecomposition on the Laplacian matrix to extract the eigenvectors corresponding to the k smallest eigenvalues. These eigenvectors form the low-dimensional representation of the data. MATLAB's eigs() function is particularly useful for sparse matrix computations. Clustering Module: Applies traditional clustering (typically K-means) to the data points in the eigenvector space to obtain final clustering results. This can be implemented using MATLAB's kmeans() function with appropriate initialization strategies. Visualization Module: Provides visualization capabilities for clustering results, displaying cluster patterns in either the original data space or feature space. MATLAB's plotting functions (scatter, plot, etc.) can be utilized with color coding for different clusters. During implementation, special attention should be paid to computational efficiency and numerical stability, especially for large-scale datasets where sparse matrix representations and optimized computation techniques may be necessary. The toolbox should feature user-friendly interfaces allowing easy module invocation and parameter configuration through function inputs or configuration structures.