# Algebraic Clustering of Affine Subspaces

arXiv: https://arxiv.org/abs/1509.06729 · HTML: https://mlsanity.com/paper/1509.06729/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2015.

## Built on by (most cited first, 3 of 3)

- [Exploring a Principled Framework for Deep Subspace Clustering](https://mlsanity.com/paper/2503.17288/) (2025)
- [Detecting adversaries in Crowdsourcing](https://mlsanity.com/paper/2110.04117/) (2021)
- [Initialization of Latent Space Coordinates via Random Linear Projections for Learning Robotic Sensory-Motor Sequences](https://mlsanity.com/paper/2202.13057/) (2022)

## Abstract (arXiv)

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.