Robust and Efficient Computation of Eigenvectors in a Generalized Spectral Method for Constrained Clustering
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:757-766, 2017.
FAST-GE is a generalized spectral method for constrained clustering [Cucuringu et al., AISTATS 2016]. It incorporates the must-link and cannot-link constraints into two Laplacian matrices and then minimizes a Rayleigh quotient via solving a generalized eigenproblem, and is considered to be simple and scalable. However, there are two unsolved issues. Theoretically, since both Laplacian matrices are positive semi-definite and the corresponding pencil is singular, it is not proven whether the minimum of the Rayleigh quotient exists and is equivalent to an eigenproblem. Computationally, the locally optimal block preconditioned conjugate gradient (LOBPCG) method is not designed for solving the eigenproblem of a singular pencil. In fact, to the best of our knowledge, there is no existing eigensolver that is immediately applicable. In this paper, we provide solutions to these two critical issues. We prove a generalization of Courant-Fischer variational principle for the Laplacian singular pencil. We propose a regularization for the pencil so that LOBPCG is applicable. We demonstrate the robustness and efficiency of proposed solutions for constrained image segmentation. The proposed theoretical and computational solutions can be applied to eigenproblems of positive semi-definite pencils arising in other machine learning algorithms, such as generalized linear discriminant analysis in dimension reduction and multisurface classification via eigenvectors.