论文标题
仿射子空间群集需要仿射约束吗？
Is an Affine Constraint Needed for Affine Subspace Clustering?
论文作者
论文摘要
基于每个数据点作为其他数据点的线性组合表达每个数据点的子空间聚类方法在计算机视觉应用程序（例如运动分割，面部和数字聚类）中取得了巨大成功。在面部聚类中，子空间是线性的，可以直接应用子空间聚类方法。在运动分段中，子空间是仿射的，并且通常会强制对系数的额外约束。但是，由于仿射子空间始终可以嵌入一个额外维度的线性子空间中，因此尚不清楚仿射约束是否真的必要。本文在理论上和经验上都表明，当环境空间的维度相对于仿射子空间的尺寸之和时，仿射约束对聚类性能的影响可忽略不计。具体而言，我们的分析提供了保证具有和不具有仿射约束的仿射子空间聚类方法的正确性的条件，并表明这些条件对高维数据满足。我们的分析的基础是亲密独立的子空间的概念，不仅提供了几何解释的正确性条件，而且还阐明了现有的仿射子空间群集的关系。
Subspace clustering methods based on expressing each data point as a linear combination of other data points have achieved great success in computer vision applications such as motion segmentation, face and digit clustering. In face clustering, the subspaces are linear and subspace clustering methods can be applied directly. In motion segmentation, the subspaces are affine and an additional affine constraint on the coefficients is often enforced. However, since affine subspaces can always be embedded into linear subspaces of one extra dimension, it is unclear if the affine constraint is really necessary. This paper shows, both theoretically and empirically, that when the dimension of the ambient space is high relative to the sum of the dimensions of the affine subspaces, the affine constraint has a negligible effect on clustering performance. Specifically, our analysis provides conditions that guarantee the correctness of affine subspace clustering methods both with and without the affine constraint, and shows that these conditions are satisfied for high-dimensional data. Underlying our analysis is the notion of affinely independent subspaces, which not only provides geometrically interpretable correctness conditions, but also clarifies the relationships between existing results for affine subspace clustering.
