IEEE Trans Image Process. 2017 Jun;26(6):2988-3001. doi: 10.1109/TIP.2017.2691557. Epub 2017 Apr 6.
Subspace clustering refers to the problem of segmenting data drawn from a union of subspaces. State-of-the-art approaches for solving this problem follow a two-stage approach. In the first step, an affinity matrix is learned from the data using sparse or low-rank minimization techniques. In the second step, the segmentation is found by applying spectral clustering to this affinity. While this approach has led to the state-of-the-art results in many applications, it is suboptimal, because it does not exploit the fact that the affinity and the segmentation depend on each other. In this paper, we propose a joint optimization framework - Structured Sparse Subspace Clustering (SC) - for learning both the affinity and the segmentation. The proposed SC framework is based on expressing each data point as a structured sparse linear combination of all other data points, where the structure is induced by a norm that depends on the unknown segmentation. Moreover, we extend the proposed SC framework into Constrained SC (CSC) in which available partial side-information is incorporated into the stage of learning the affinity. We show that both the structured sparse representation and the segmentation can be found via a combination of an alternating direction method of multipliers with spectral clustering. Experiments on a synthetic data set, the Extended Yale B face data set, the Hopkins 155 motion segmentation database, and three cancer data sets demonstrate the effectiveness of our approach.
子空间聚类是指将数据从子空间的并集中分割出来的问题。解决这个问题的最新方法采用两阶段的方法。在第一步中，使用稀疏或低秩最小化技术从数据中学习相似矩阵。在第二步中，通过对这个相似矩阵应用谱聚类来找到分割。虽然这种方法在许多应用中取得了最先进的结果，但它并不理想，因为它没有利用相似矩阵和分割相互依赖的事实。在本文中，我们提出了一种联合优化框架——结构化稀疏子空间聚类（SC），用于学习相似矩阵和分割。所提出的 SC 框架基于将每个数据点表示为所有其他数据点的结构化稀疏线性组合，其中结构是由依赖于未知分割的范数诱导的。此外，我们将所提出的 SC 框架扩展为约束 SC（CSC），其中将可用的部分边信息合并到学习相似矩阵的阶段中。我们表明，通过交替方向乘子法与谱聚类的组合，可以找到结构化稀疏表示和分割。在合成数据集、扩展耶鲁 B 人脸数据集、霍普金斯 155 运动分割数据库和三个癌症数据集上的实验表明了我们方法的有效性。
