K-Way Spectral Clustering

In spectral clustering (Luxburg, 2007), the dataset is represented as a similarity graph G = (V, E). The vertices represent the data points. Two vertices are connected if the similarity between the corresponding data points is larger than a certain threshold, and the edge is weighted by the similarity value. Clustering is achieved by choosing a suitable partition of the graph that each group corresponds to one cluster.

A good partition (i.e., a good clustering) is that the edges between different groups have overall low weights and the edges within a group have high weights, which indicates that the points in different clusters are dissimilar from each other and the points within the same cluster are similar to each other. One basic spectral clustering algorithm finds a good partition in the following way:

Given a set of data points P and the similarity matrix S, where S _ij measures the similarity between points i, j ∈ P, form a graph. Build a Laplacian matrix L of the graph,

$$L = I -...