# Is an Affine Constraint Needed for Affine Subspace Clustering?

arXiv: https://arxiv.org/abs/2005.03888 · HTML: https://mlsanity.com/paper/2005.03888/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2020.

## Built on by (most cited first, 10 of 10)

- [Learning a Self-Expressive Network for Subspace Clustering](https://mlsanity.com/paper/2110.04318/) (2021)
- [Unsupervised Manifold Linearizing and Clustering](https://mlsanity.com/paper/2301.01805/) (2023)
- [Adaptive Attribute and Structure Subspace Clustering Network](https://mlsanity.com/paper/2109.13742/) (2021)
- [Learning idempotent representation for subspace clustering](https://mlsanity.com/paper/2207.14431/) (2022)
- [Noisy ℓ0-Sparse Subspace Clustering on Dimensionality Reduced Data](https://mlsanity.com/paper/2206.11079/) (2022)
- [Revisiting data augmentation for subspace clustering](https://mlsanity.com/paper/2207.09728/) (2022)
- [Subspace Clustering in Wavelet Packets Domain](https://mlsanity.com/paper/2406.03819/) (2024)
- [Label-independent hyperparameter-free self-supervised single-view deep subspace clustering](https://mlsanity.com/paper/2504.18179/) (2025)
- [Affine Subspace Models and Clustering for Patch-Based Image Denoising](https://mlsanity.com/paper/2512.07259/) (2025)
- [Nonlinear mixture model motivated subspace clustering](https://mlsanity.com/paper/2606.29261/) (2026)

## Abstract (arXiv)

Subspace clustering methods based on expressing each data point as a linear combination of other data points have achieved great success in computer vision applications such as motion segmentation, face and digit clustering. In face clustering, the subspaces are linear and subspace clustering methods can be applied directly. In motion segmentation, the subspaces are affine and an additional affine constraint on the coefficients is often enforced. However, since affine subspaces can always be embedded into linear subspaces of one extra dimension, it is unclear if the affine constraint is really necessary. This paper shows, both theoretically and empirically, that when the dimension of the ambient space is high relative to the sum of the dimensions of the affine subspaces, the affine constraint has a negligible effect on clustering performance. Specifically, our analysis provides conditions that guarantee the correctness of affine subspace clustering methods both with and without the affine constraint, and shows that these conditions are satisfied for high-dimensional data. Underlying our analysis is the notion of affinely independent subspaces, which not only provides geometrically interpretable correctness conditions, but also clarifies the relationships between existing results for affine subspace clustering.