Grassmann流形上奇异值表示的部分和极小化
Partial Sum Minimization of Singular Values Representation on Grassmann Manifolds
摘要：
低秩表示作为一种重要的子空间聚类方法，近年来受到了广泛的关注。为了进一步提高LRR的性能，扩大其应用范围，还有几个问题需要解决。LRR中的核范数没有充分利用许多实际问题中已知的秩的先验知识。LRR是为线性空间中的向量数据设计的，因此不适用于具有内在非线性流形结构的高维数据。本文提出了一种流形值Grassmann数据的扩展LRR模型，该模型通过最小化奇异值的部分和而不是核范数来融合先验知识，即奇异值表示的部分和最小化（Partial Sum Minimization of Singular Values Representation，GPSSVR）。新模型不仅在低秩时加强了数据的全局结构，而且通过最小化较小的奇异值来保留重要信息。为了进一步保持格拉斯曼点之间的局部结构，我们还将拉普拉斯惩罚与GPSSVR相结合。在GPSSVR模型的基础上，提出了一种有效的求解优化问题的算法。在一些广泛使用的人体动作视频数据集和真实场景数据集上对所提出的模型和算法进行了评估。实验结果表明，该方法明显优于其他国家的最先进的方法。
英文摘要：
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.
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