Spectral Clustering Explained: How Eigenvectors Reveal Complex Cluster Structures

Spectral clustering is a graph-based clustering method that outperforms K-means on non-linear, complex cluster structures like the two-moon dataset. It works by building a similarity matrix using a Gaussian kernel, constructing the graph Laplacian matrix, then performing eigendecomposition to find eigenvectors that embed data into a lower-dimensional space where clusters become linearly separable. K-means is then applied in this new spectral embedding space. The post walks through each step from scratch in Python using NumPy and scikit-learn, covering the similarity matrix, degree matrix, Laplacian construction, eigengap heuristic for choosing the number of clusters, and gamma hyperparameter tuning.