# Oracle Based Active Set Algorithm for Scalable Elastic Net Subspace Clustering

arXiv: https://arxiv.org/abs/1605.02633 · HTML: https://mlsanity.com/paper/1605.02633/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 May 2016.

## Built on by (most cited first, 30 of 40)

- [Multi-dimensional concept discovery (MCD): A unifying framework with completeness guarantees](https://mlsanity.com/paper/2301.11911/) (2023)
- [Neural Manifold Clustering and Embedding](https://mlsanity.com/paper/2201.10000/) (2022)
- [Learning a Self-Expressive Network for Subspace Clustering](https://mlsanity.com/paper/2110.04318/) (2021)
- [Image Clustering via the Principle of Rate Reduction in the Age of Pretrained Models](https://mlsanity.com/paper/2306.05272/) (2023)
- [PaCE: Parsimonious Concept Engineering for Large Language Models](https://mlsanity.com/paper/2406.04331/) (2024)
- [Towards Clustering-friendly Representations: Subspace Clustering via Graph Filtering](https://mlsanity.com/paper/2106.09874/) (2021)
- [Unsupervised Manifold Linearizing and Clustering](https://mlsanity.com/paper/2301.01805/) (2023)
- [A Simple Approach to Automated Spectral Clustering](https://mlsanity.com/paper/2107.12183/) (2021)
- [Adaptive Attribute and Structure Subspace Clustering Network](https://mlsanity.com/paper/2109.13742/) (2021)
- [Exploring a Principled Framework for Deep Subspace Clustering](https://mlsanity.com/paper/2503.17288/) (2025)
- [Sparse Subspace Clustering for Concept Discovery (SSCCD)](https://mlsanity.com/paper/2203.06043/) (2022)
- [Concept Lancet: Image Editing with Compositional Representation Transplant](https://mlsanity.com/paper/2504.02828/) (2025)
- [Deep Double Self-Expressive Subspace Clustering](https://mlsanity.com/paper/2306.11592/) (2023)
- [Learning idempotent representation for subspace clustering](https://mlsanity.com/paper/2207.14431/) (2022)
- [Neural Normalized Cut: A Differential and Generalizable Approach for Spectral Clustering](https://mlsanity.com/paper/2503.09260/) (2025)
- [Provable Data Clustering via Innovation Search](https://mlsanity.com/paper/2108.06888/) (2021)
- [Detecting adversaries in Crowdsourcing](https://mlsanity.com/paper/2110.04117/) (2021)
- [PMSSC: Parallelizable multi-subset based self-expressive model for subspace clustering](https://mlsanity.com/paper/2111.12232/) (2021)
- [Greedier is Better: Selecting Multiple Neighbors per Iteration for Sparse Subspace Clustering](https://mlsanity.com/paper/2204.02572/) (2022)
- [Density peak clustering using tensor network](https://mlsanity.com/paper/2302.00192/) (2023)
- [Deep Structure and Attention Aware Subspace Clustering](https://mlsanity.com/paper/2312.15577/) (2023)
- [Graph Cuts with Arbitrary Size Constraints Through Optimal Transport](https://mlsanity.com/paper/2402.04732/) (2024)
- [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)
- [Beyond Linear Subspace Clustering: A Comparative Study of Nonlinear Manifold Clustering Algorithms](https://mlsanity.com/paper/2103.10656/) (2021)
- [Learning Deep Representation with Energy-Based Self-Expressiveness for Subspace Clustering](https://mlsanity.com/paper/2110.15037/) (2021)
- [Dictionary learning for clustering on hyperspectral images](https://mlsanity.com/paper/2202.00990/) (2022)
- [Self-Supervised Deep Subspace Clustering with Entropy-norm](https://mlsanity.com/paper/2206.04958/) (2022)
- [Revisiting data augmentation for subspace clustering](https://mlsanity.com/paper/2207.09728/) (2022)
- [Inv-SENnet: Invariant Self Expression Network for clustering under biased data](https://mlsanity.com/paper/2211.06780/) (2022)

## Abstract (arXiv)

State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with ℓ1, ℓ2 or nuclear norms. ℓ1 regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. ℓ2 and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed ℓ1, ℓ2 and nuclear norm regularizations offer a balance between the subspace-preserving and connectedness properties, but this comes at the cost of increased computational complexity. This paper studies the geometry of the elastic net regularizer (a mixture of the ℓ1 and ℓ2 norms) and uses it to derive a provably correct and scalable active set method for finding the optimal coefficients. Our geometric analysis also provides a theoretical justification and a geometric interpretation for the balance between the connectedness (due to ℓ2 regularization) and subspace-preserving (due to ℓ1 regularization) properties for elastic net subspace clustering. Our experiments show that the proposed active set method not only achieves state-of-the-art clustering performance, but also efficiently handles large-scale datasets.