Constrained Spectral Clustering using L1 Regularization

Abstract

Constrained spectral clustering is a semi-supervised learning problem that aims at incorporating userdefined constraints in spectral clustering. Typically, there are two kinds of constraints: (i) must-link, and (ii) cannot-link. These constraints represent prior knowledge indicating whether two data objects should be in the same cluster or not; thereby aiding in clustering. In this paper, we propose a novel approach that uses convex subproblems to incorporate constraints in spectral clustering and co-clustering. In comparison to the prior state-of-art approaches, our approach presents a more natural way to incorporate constraints in the spectral methods and allows us to make a trade off between the number of satisfied constraints and the quality of partitions on the original graph. We use an L1 regularizer analogous to LASSO, often used in literature to induce sparsity, in order to control the number of constraints satisfied. Our approach can handle both must-link and cannot-link constraints, unlike a large number of previous approaches that mainly work on the former. Further, our formulation is based on the reduction to a convex subproblem which is relatively easy to solve using existing solvers. We test our proposed approach on real world datasets and show its effectiveness for both spectral clustering and co-clustering over the prior state-of-art.