Algebraic Clustering of Affine Subspaces

TL;DR

This paper extends algebraic subspace clustering (ASC) to affine subspaces by proving that the homogenization technique preserves key geometric properties, ensuring ASC's correctness for affine cases.

Contribution

It provides a rigorous algebraic geometric analysis demonstrating that ASC is valid for affine subspaces using the homogenization approach.

Findings

Homogenization preserves general position of points.

Transversality of the union of subspaces is maintained.

ASC is guaranteed to correctly cluster affine subspaces.

Abstract

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.