What is the role of eigen gap in PCA and spectral methods?

Eigen gap helps determine the optimal number of dimensions by identifying sharp drops in eigenvalues.

The eigen gap is the difference between consecutive eigenvalues. A large gap indicates that the previous components capture most of the meaningful variance or structure. In PCA, it helps decide how many principal components to retain. In spectral clustering, it helps determine the number of clusters by analyzing the Laplacian spectrum.