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Spectral clustering is one of the fundamental unsupervised learning methodswidely used in data analysis. Sparse spectral clustering (SSC) imposes sparsityto the spectral clustering and it improves the interpretability of the model.This paper considers a widely adopted model for SSC, which can be formulated asan optimization problem over the Stiefel manifold with nonsmooth and nonconvexobjective. Such an optimization problem is very challenging to solve. Existingmethods usually solve its convex relaxation or need to smooth its nonsmoothpart using certain smoothing techniques. In this paper, we propose a manifoldproximal linear method (ManPL) that solves the original SSC formulation. Wealso extend the algorithm to solve the multiple-kernel SSC problems, for whichan alternating ManPL algorithm is proposed. Convergence and iterationcomplexity results of the proposed methods are established. We demonstrate theadvantage of our proposed methods over existing methods via the single-cell RNAsequencing data analysis.

A Manifold Proximal Linear Method for Sparse Spectral Clustering with Application to Single-Cell RNA Sequencing Data Analysis