On Constrained Spectral Clustering and Its Applications

TL;DR

This paper introduces a flexible, principled framework for constrained spectral clustering that explicitly encodes constraints, guarantees constraint satisfaction levels, and can be solved efficiently, with validation on artificial and real datasets.

Contribution

It presents a novel formulation for constrained spectral clustering that explicitly incorporates constraints into a polynomial-time solvable optimization problem, improving over previous implicit methods.

Findings

Effective constraint encoding with belief degrees

Guarantees constraint satisfaction thresholds

Validated on artificial and real datasets

Abstract

Constrained clustering has been well-studied for algorithms such as $K$-means and hierarchical clustering. However, how to satisfy many constraints in these algorithmic settings has been shown to be intractable. One alternative to encode many constraints is to use spectral clustering, which remains a developing area. In this paper, we propose a flexible framework for constrained spectral clustering. In contrast to some previous efforts that implicitly encode Must-Link and Cannot-Link constraints by modifying the graph Laplacian or constraining the underlying eigenspace, we present a more natural and principled formulation, which explicitly encodes the constraints as part of a constrained optimization problem. Our method offers several practical advantages: it can encode the degree of belief in Must-Link and Cannot-Link constraints; it guarantees to lower-bound how well the given constraints are satisfied using a user-specified threshold; it can be solved deterministically in polynomial time through generalized eigendecomposition. Furthermore, by inheriting the objective function from spectral clustering and encoding the constraints explicitly, much of the existing analysis of unconstrained spectral clustering techniques remains valid for our formulation. We validate the effectiveness of our approach by empirical results on both artificial and real datasets. We also demonstrate an innovative use of encoding large number of constraints: transfer learning via constraints.