$K$-way $p$-spectral clustering on Grassmann manifolds

Spectral methods have gained a lot of recent attention due to the simplicity of their implementation and their solid mathematical background. We revisit spectral graph clustering, and reformulate in the $p$-norm the continuous problem of minimizing the graph Laplacian Rayleigh quotient. The value of $p \in (1,2]$ is reduced, promoting sparser solution vectors that correspond to optimal clusters as $p$ approaches one. The computation of multiple $p$-eigenvectors of the graph $p$-Laplacian, a nonlinear generalization of the standard graph Laplacian, is achieved by the minimization of our objective function on the Grassmann manifold, hence ensuring the enforcement of the orthogonality constraint between them. Our approach attempts to bridge the fields of graph clustering and nonlinear numerical optimization, and employs a robust algorithm to obtain clusters of high quality. The benefits of the suggested method are demonstrated in a plethora of artificial and real-world graphs. Our results are compared against standard spectral clustering methods and the current state-of-the-art algorithm for clustering using the graph $p$-Laplacian variant.