A Convex Relaxation Approach to the Affine Subspace Clustering Problem

Abstract. Prototypical data clustering is known to su↵er from poor initializations. Recently, a semidefinite relaxation has been proposed to overcome this issue and to enable the use of convex programming instead of ad-hoc procedures. Unfortunately, this relaxation does not extend to the more involved case where clusters are defined by parametric models, and where the computation of means has to be replaced by parametric regression. In this paper, we provide a novel convex relaxation approach to this more involved problem class that is relevant to many scenarios of unsupervised data analysis. Our approach applies, in particular, to data sets where assumptions of model recovery through sparse regularization, like the independent subspace model, do not hold. Our mathematical analysis enables to distinguish scenarios where the relaxation is tight enough and scenarios where the approach breaks down.