# Doubly Stochastic Subspace Clustering

arXiv: https://arxiv.org/abs/2011.14859 · HTML: https://mlsanity.com/paper/2011.14859/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2020.

## Built on by (most cited first, 14 of 14)

- [Image Clustering via the Principle of Rate Reduction in the Age of Pretrained Models](https://mlsanity.com/paper/2306.05272/) (2023)
- [Unsupervised Manifold Linearizing and Clustering](https://mlsanity.com/paper/2301.01805/) (2023)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling](https://mlsanity.com/paper/2209.08004/) (2022)
- [Exploring a Principled Framework for Deep Subspace Clustering](https://mlsanity.com/paper/2503.17288/) (2025)
- [Quantum Doubly Stochastic Transformers](https://mlsanity.com/paper/2504.16275/) (2025)
- [Adaptive Graph Convolutional Subspace Clustering](https://mlsanity.com/paper/2305.03414/) (2023)
- [Learning idempotent representation for subspace clustering](https://mlsanity.com/paper/2207.14431/) (2022)
- [Manifold Learning with Sparse Regularised Optimal Transport](https://mlsanity.com/paper/2307.09816/) (2023)
- [ALPCAHUS: Subspace Clustering for Heteroscedastic Data](https://mlsanity.com/paper/2505.18918/) (2025)
- [fMBN-E: Efficient Unsupervised Network Structure Ensemble and Selection for Clustering](https://mlsanity.com/paper/2107.02071/) (2021)
- [Graph Cut-guided Maximal Coding Rate Reduction for Learning Image Embedding and Clustering](https://mlsanity.com/paper/2412.18930/) (2024)
- [Scalable Deep Subspace Clustering Network](https://mlsanity.com/paper/2512.21434/) (2025)
- [LoomSC: Scalable Deep Subspace Clustering with Projector Factorization and Exact Spectral Reduction](https://mlsanity.com/paper/2610.10266/) (2026)

## Abstract (arXiv)

Many state-of-the-art subspace clustering methods follow a two-step process by first constructing an affinity matrix between data points and then applying spectral clustering to this affinity. Most of the research into these methods focuses on the first step of generating the affinity, which often exploits the self-expressive property of linear subspaces, with little consideration typically given to the spectral clustering step that produces the final clustering. Moreover, existing methods often obtain the final affinity that is used in the spectral clustering step by applying ad-hoc or arbitrarily chosen postprocessing steps to the affinity generated by a self-expressive clustering formulation, which can have a significant impact on the overall clustering performance. In this work, we unify these two steps by learning both a self-expressive representation of the data and an affinity matrix that is well-normalized for spectral clustering. In our proposed models, we constrain the affinity matrix to be doubly stochastic, which results in a principled method for affinity matrix normalization while also exploiting known benefits of doubly stochastic normalization in spectral clustering. We develop a general framework and derive two models: one that jointly learns the self-expressive representation along with the doubly stochastic affinity, and one that sequentially solves for one then the other. Furthermore, we leverage sparsity in the problem to develop a fast active-set method for the sequential solver that enables efficient computation on large datasets. Experiments show that our method achieves state-of-the-art subspace clustering performance on many common datasets in computer vision.