Table 1 Procedure for Spectral Clustering.

1.

Compute the correlation ρ _ij between all pairs of n data points i and j.

2.

Form the similarity matrix S ∈ ℝ^n×ndefined by s _ij = exp [- sin² (arccos(ρ _ij )/2)/σ²], where σ is a scaling parameter (σ = 1 in the reported results).

3.

Define D to be the diagonal matrix whose (i,i) elements are the column sums of S.

4.

Define the Laplacian L = I - D^-1/2SD^-1/2.

5.

Find the eigenvectors {v₀, v₁, v₂, . . . , v_n-1} with corresponding eigenvalues 0 ≤ λ₁ ≤ λ₂ ≤ ⋯ ≤ λ_n-1of L.

6.

Determine from the eigendecomposition the optimal dimensionality l and natural number of clusters k (see text).

7.

Construct the embedded data by using the first l eigenvectors to provide coordinates for the data (i.e., sample i is assigned to the point in the Laplacian eigenspace with coordinates given by the i th entries of each of the first l eigenvectors, similar to PCA).

8.

Using k-means, cluster the l-dimensional embedded data into k clusters.