Partial Sum Minimization of Singular Values Representation on Grassmann Manifolds

Abstract

Clustering is one of the fundamental topics in data mining and pattern recognition. As a prospective clustering method, the subspace clustering has made considerable progress in recent researches, e.g., sparse subspace clustering (SSC) and low rank representation (LRR). However, most existing subspace clustering algorithms are designed for vectorial data from linear spaces, thus not suitable for high dimensional data with intrinsic non-linear manifold structure. For high dimensional or manifold data, few research pays attention to clustering problems. The purpose of clustering on manifolds tends to cluster manifold-valued data into several groups according to the mainfold-based similarity metric. This paper proposes an extended LRR model for manifold-valued Grassmann data which incorporates prior knowledge by minimizing partial sum of singular values instead of the nuclear norm, namely Partial Sum minimization of Singular Values Representation (GPSSVR). The new model not only enforces the global structure of data in low rank, but also retains important information by minimizing only smaller singular values. To further maintain the local structures among Grassmann points, we also integrate the Laplacian penalty with GPSSVR. The proposed model and algorithms are assessed on a public human face dataset, some widely used human action video datasets and a real scenery dataset. The experimental results show that the proposed methods obviously outperform other state-of-the-art methods.