# Structured Sparse Subspace Clustering: A Joint Affinity Learning and Subspace Clustering Framework

arXiv: https://arxiv.org/abs/1610.05211 · HTML: https://mlsanity.com/paper/1610.05211/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2016.

## Built on by (most cited first, 26 of 26)

- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [Structured Graph Learning for Scalable Subspace Clustering: From Single-view to Multi-view](https://mlsanity.com/paper/2102.07943/) (2021)
- [Learning a Self-Expressive Network for Subspace Clustering](https://mlsanity.com/paper/2110.04318/) (2021)
- [Towards Clustering-friendly Representations: Subspace Clustering via Graph Filtering](https://mlsanity.com/paper/2106.09874/) (2021)
- [Exploring a Principled Framework for Deep Subspace Clustering](https://mlsanity.com/paper/2503.17288/) (2025)
- [Enhanced Latent Multi-view Subspace Clustering](https://mlsanity.com/paper/2312.14763/) (2023)
- [Nonlinear subspace clustering by functional link neural networks](https://mlsanity.com/paper/2402.02051/) (2024)
- [Structuring Sparsity: Block-Sparse Featurizers Capture Visual Concept Manifolds](https://mlsanity.com/paper/2606.25234/) (2026)
- [Self-supervised Symmetric Nonnegative Matrix Factorization](https://mlsanity.com/paper/2103.01689/) (2021)
- [Learning idempotent representation for subspace clustering](https://mlsanity.com/paper/2207.14431/) (2022)
- [Greedier is Better: Selecting Multiple Neighbors per Iteration for Sparse Subspace Clustering](https://mlsanity.com/paper/2204.02572/) (2022)
- [Enriched Robust Multi-View Kernel Subspace Clustering](https://mlsanity.com/paper/2205.10495/) (2022)
- [Temporal Rate Reduction Clustering for Human Motion Segmentation](https://mlsanity.com/paper/2506.21249/) (2025)
- [Subspace Clustering of Subspaces: Unifying Canonical Correlation Analysis and Subspace Clustering](https://mlsanity.com/paper/2509.18653/) (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)
- [Neural Ordinary Differential Equation Model for Evolutionary Subspace Clustering and Its Applications](https://mlsanity.com/paper/2107.10484/) (2021)
- [Rectified Euler k-means and Beyond](https://mlsanity.com/paper/2108.03081/) (2021)
- [Learning Deep Representation with Energy-Based Self-Expressiveness for Subspace Clustering](https://mlsanity.com/paper/2110.15037/) (2021)
- [Fusion Subspace Clustering for Incomplete Data](https://mlsanity.com/paper/2205.10872/) (2022)
- [One-step Bipartite Graph Cut: A Normalized Formulation and Its Application to Scalable Subspace Clustering](https://mlsanity.com/paper/2305.07386/) (2023)
- [Semi-supervised segmentation of land cover images using nonlinear canonical correlation analysis with multiple features and t-SNE](https://mlsanity.com/paper/2401.12164/) (2024)
- [Masked Subspace Clustering Methods](https://mlsanity.com/paper/2505.06863/) (2025)
- [CoSeNet: A Novel Approach for Optimal Segmentation of Correlation Matrices](https://mlsanity.com/paper/2512.21000/) (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)

Subspace clustering refers to the problem of segmenting data drawn from a union of subspaces. State-of-the-art approaches for solving this problem follow a two-stage approach. In the first step, an affinity matrix is learned from the data using sparse or low-rank minimization techniques. In the second step, the segmentation is found by applying spectral clustering to this affinity. While this approach has led to state-of-the-art results in many applications, it is sub-optimal because it does not exploit the fact that the affinity and the segmentation depend on each other. In this paper, we propose a joint optimization framework --- Structured Sparse Subspace Clustering (S3C) --- for learning both the affinity and the segmentation. The proposed S3C framework is based on expressing each data point as a structured sparse linear combination of all other data points, where the structure is induced by a norm that depends on the unknown segmentation. Moreover, we extend the proposed S3C framework into Constrained Structured Sparse Subspace Clustering (CS3C) in which available partial side-information is incorporated into the stage of learning the affinity. We show that both the structured sparse representation and the segmentation can be found via a combination of an alternating direction method of multipliers with spectral clustering. Experiments on a synthetic data set, the Extended Yale B data set, the Hopkins 155 motion segmentation database, and three cancer data sets demonstrate the effectiveness of our approach.