Atomic Recovery Property for Multi-view Subspace-Preserving Recovery

Abstract

As the theoretical underpinnings for subspace clustering and classification, subspace-preserving recovery has attracted intensive attention in recent years. However, previous theoretical advances for subspace-preserving recovery only focus on the single-view data and most of them are based on conditions that are only sufficient. In this paper, we propose a necessary and sufficient condition referred to as Atomic Recovery Property (ARP) for multi-view subspace-preserving recovery. To this end, we generalize the atomic norm from single-view data to multi-view data and define the Multi-view Atomic Norm (MAN). Our another contribution is to provide a geometrically more interpretable characterization of ARP with respect to the unit ball of MAN. Based on the proposed multi-view subspace-preserving recovery theory, we also derive novel theoretical results for multi-view subspace clustering and classification, respectively.