Partial Sum Minimization of Singular Values Representation on Grassmann Manifolds

TL;DR

This paper introduces GPSSVR, a novel low-rank representation method for Grassmann manifold data that improves clustering performance by selectively minimizing smaller singular values and preserving local structures.

Contribution

It extends low rank representation to manifold-valued data by minimizing partial singular values and incorporates local structure preservation via Laplacian penalty.

Findings

Outperforms state-of-the-art methods on human action video datasets

Effectively preserves local and global data structures

Demonstrates robustness on real scenery dataset

Abstract

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.