Algebraic Clustering of Affine Subspaces

Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work studied this problem in the case of linear subspaces using algebraic, iterative, statistical, low-rank and sparse representation techniques. While such methods have been applied to the case of affine subspaces, the theory of affine subspace clustering remains largely unstudied. This paper rigorously extends the theory of algebraic subspace clustering to the case of affine subspaces, which naturally arise in practical applications. We model a union of affine subspaces as an affine variety whose irreducible components give the individual subspaces. This leads to a decomposition method which, through a process called projectivization, is shown to be equivalent to the classical algebraic method for linear subspaces.