# On Geometric Analysis of Affine Sparse Subspace Clustering

arXiv: https://arxiv.org/abs/1808.05965 · HTML: https://mlsanity.com/paper/1808.05965/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Aug 2018.

## Built on by (most cited first, 12 of 12)

- [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)
- [Adversarial Examples Might be Avoidable: The Role of Data Concentration in Adversarial Robustness](https://mlsanity.com/paper/2309.16096/) (2023)
- [Exploring a Principled Framework for Deep Subspace Clustering](https://mlsanity.com/paper/2503.17288/) (2025)
- [Deep Adversarial Inconsistent Cognitive Sampling for Multi-view Progressive Subspace Clustering](https://mlsanity.com/paper/2101.03783/) (2021)
- [Weighted Sparse Subspace Representation: A Unified Framework for Subspace Clustering, Constrained Clustering, and Active Learning](https://mlsanity.com/paper/2106.04330/) (2021)
- [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)
- [Affine Subspace Models and Clustering for Patch-Based Image Denoising](https://mlsanity.com/paper/2512.07259/) (2025)
- [Modeling high dimensional point clouds with the spherical cluster model](https://mlsanity.com/paper/2512.21960/) (2025)
- [Jointly Learning Structured Representations and Stabilized Affinity for Human Motion Segmentation](https://mlsanity.com/paper/2605.05753/) (2026)
- [Learning Deep Modality-Shared Self-Expressiveness for Image Clustering with Textual Information](https://mlsanity.com/paper/2608.08418/) (2026)

## Abstract (arXiv)

Sparse subspace clustering (SSC) is a state-of-the-art method for segmenting a set of data points drawn from a union of subspaces into their respective subspaces. It is now well understood that SSC produces subspace-preserving data affinity under broad geometric conditions but suffers from a connectivity issue. In this paper, we develop a novel geometric analysis for a variant of SSC, named affine SSC (ASSC), for the problem of clustering data from a union of affine subspaces. Our contributions include a new concept called affine independence for capturing the arrangement of a collection of affine subspaces. Under the affine independence assumption, we show that ASSC is guaranteed to produce subspace-preserving affinity. Moreover, inspired by the phenomenon that the ℓ1 regularization no longer induces sparsity when the solution is nonnegative, we further show that subspace-preserving recovery can be achieved under much weaker conditions for all data points other than the extreme points of samples from each subspace. In addition, we confirm a curious observation that the affinity produced by ASSC may be subspace-dense---which could guarantee the subspace-preserving affinity of ASSC to produce correct clustering under rather weak conditions. We validate the theoretical findings on carefully designed synthetic data and evaluate the performance of ASSC on several real data sets.