Algebraic Clustering of Affine Subspaces
This provides a theoretical foundation for affine subspace clustering, addressing a gap in prior work that only covered linear subspaces, which is incremental but important for applications in computer vision and pattern recognition.
The paper tackles the problem of subspace clustering for affine subspaces by proving that the homogenization trick preserves key geometric properties, establishing the correctness of Algebraic Subspace Clustering (ASC) for affine subspaces.
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 subpaces.