Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal. In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces.

中文翻译：

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仿射子空间的代数聚类

子空间聚类是机器学习中的一个重要问题，在计算机视觉和模式识别中有许多应用。先前的工作已经使用代数，迭代，统计，低秩和稀疏表示技术研究了这个问题。尽管这些方法已应用于线性子空间和仿射子空间，但仅在线性子空间的情况下才建立了理论结果。例如，当数据点位于以下位置时，可以保证代数子空间聚类（ASC）提供正确的聚类一般职位 子空间的并集是 横向的。在本文中，我们以仿射子空间的严格方式研究了ASC的性质。利用代数几何的概念，我们证明了均质化技巧 将仿射子空间并集中的点嵌入线性子空间并集中的点，保留了嵌入空间中点的一般位置和子空间并集的横向性，从而建立了仿射子空间中ASC的正确性。