Algebraic Clustering of 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.

References in corpus (3)

• A geometric analysis of subspace clustering with outliers
• Robust subspace clustering
• Castelnuovo-Mumford regularity of products of ideals

Cited by in corpus (9)

• Structured Sparse Subspace Clustering: A Joint Affinity Learning and Subspace Clustering Framework
• On Geometric Analysis of Affine Sparse Subspace Clustering
• Homomorphic Sensing of Subspace Arrangements
• Restricted Connection Orthogonal Matching Pursuit For Sparse Subspace Clustering
• Beyond Linear Subspace Clustering: A Comparative Study of Nonlinear Manifold Clustering Algorithms
• Scaled Simplex Representation for Subspace Clustering
• Robust Fuzzy local k-plane clustering with mixture distance of hinge loss and L1 norm
• Recovery of surfaces and functions in high dimensions: sampling theory and links to neural networks
• Is an Affine Constraint Needed for Affine Subspace Clustering?