Partial Sum Minimization of Singular Values Representation on Grassmann Manifolds
This work addresses subspace clustering for data with non-linear manifold structures, such as video analysis, but it is incremental as it builds upon existing low rank representation methods.
The paper tackled subspace clustering for high-dimensional data on Grassmann manifolds by proposing an extended low rank representation model that minimizes partial sums of singular values and integrates a Laplacian penalty, resulting in improved performance over state-of-the-art methods on human action video and scenery datasets.
As a significant subspace clustering method, low rank representation (LRR) has attracted great attention in recent years. To further improve the performance of LRR and extend its applications, there are several issues to be resolved. The nuclear norm in LRR does not sufficiently use the prior knowledge of the rank which is known in many practical problems. The LRR is designed for vectorial data from linear spaces, thus not suitable for high dimensional data with intrinsic non-linear manifold structure. 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. An effective algorithm is proposed to solve the optimization problem based on the GPSSVR model. The proposed model and algorithms are assessed on 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.