# Neural Manifold Clustering and Embedding

arXiv: https://arxiv.org/abs/2201.10000 · HTML: https://mlsanity.com/paper/2201.10000/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 57 references that are in the map.
- First posted on arXiv 24 Jan 2022.

## Builds on (most cited first, 30 of 30)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Large Batch Training of Convolutional Networks](https://mlsanity.com/paper/1708.03888/) (2017)
- [Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss](https://mlsanity.com/paper/2106.04156/) (2021)
- [Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style](https://mlsanity.com/paper/2106.04619/) (2021)
- [Contrastive Clustering](https://mlsanity.com/paper/2009.09687/) (2020)
- [Contrastive Learning Inverts the Data Generating Process](https://mlsanity.com/paper/2102.08850/) (2021)
- [Learning Diverse and Discriminative Representations via the Principle of Maximal Coding Rate Reduction](https://mlsanity.com/paper/2006.08558/) (2020)
- [Solo-learn: A Library of Self-supervised Methods for Visual Representation Learning](https://mlsanity.com/paper/2108.01775/) (2021)
- [Deep Subspace Clustering Networks](https://mlsanity.com/paper/1709.02508/) (2017)
- [Clustering-friendly Representation Learning via Instance Discrimination and Feature Decorrelation](https://mlsanity.com/paper/2106.00131/) (2021)
- [Improving Unsupervised Image Clustering With Robust Learning](https://mlsanity.com/paper/2012.11150/) (2020)
- [MiCE: Mixture of Contrastive Experts for Unsupervised Image Clustering](https://mlsanity.com/paper/2105.01899/) (2021)
- [Oracle Based Active Set Algorithm for Scalable Elastic Net Subspace Clustering](https://mlsanity.com/paper/1605.02633/) (2016)
- [You Never Cluster Alone](https://mlsanity.com/paper/2106.01908/) (2021)
- [The Sparse Manifold Transform](https://mlsanity.com/paper/1806.08887/) (2018)
- [Self-Supervised Convolutional Subspace Clustering Network](https://mlsanity.com/paper/1905.00149/) (2019)
- [Learning a Self-Expressive Network for Subspace Clustering](https://mlsanity.com/paper/2110.04318/) (2021)
- [Clustering by Maximizing Mutual Information Across Views](https://mlsanity.com/paper/2107.11635/) (2021)
- [A Critique of Self-Expressive Deep Subspace Clustering](https://mlsanity.com/paper/2010.03697/) (2020)
- [ScatSimCLR: self-supervised contrastive learning with pretext task regularization for small-scale datasets](https://mlsanity.com/paper/2108.13939/) (2021)
- [Representation Learning for Clustering via Building Consensus](https://mlsanity.com/paper/2105.01289/) (2021)
- [LogDet Rank Minimization with Application to Subspace Clustering](https://mlsanity.com/paper/1507.00908/) (2015)

## Built on by (most cited first, 30 of 44)

- [RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank](https://mlsanity.com/paper/2210.02885/) (2022)
- [On the duality between contrastive and non-contrastive self-supervised learning](https://mlsanity.com/paper/2206.02574/) (2022)
- [Learning and Leveraging World Models in Visual Representation Learning](https://mlsanity.com/paper/2403.00504/) (2024)
- [Self-supervised learning of Split Invariant Equivariant representations](https://mlsanity.com/paper/2302.10283/) (2023)
- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [Self-Supervised Learning with Lie Symmetries for Partial Differential Equations](https://mlsanity.com/paper/2307.05432/) (2023)
- [Your Contrastive Learning Is Secretly Doing Stochastic Neighbor Embedding](https://mlsanity.com/paper/2205.14814/) (2022)
- [Image Clustering via the Principle of Rate Reduction in the Age of Pretrained Models](https://mlsanity.com/paper/2306.05272/) (2023)
- [Matrix Information Theory for Self-Supervised Learning](https://mlsanity.com/paper/2305.17326/) (2023)
- [Simplifying DINO via Coding Rate Regularization](https://mlsanity.com/paper/2502.10385/) (2025)
- [EMP-SSL: Towards Self-Supervised Learning in One Training Epoch](https://mlsanity.com/paper/2304.03977/) (2023)
- [Interpretable Deep Clustering for Tabular Data](https://mlsanity.com/paper/2306.04785/) (2023)
- [Neuromorphic Visual Scene Understanding with Resonator Networks](https://mlsanity.com/paper/2208.12880/) (2022)
- [Variance-Covariance Regularization Improves Representation Learning](https://mlsanity.com/paper/2306.13292/) (2023)
- [Unsupervised Manifold Linearizing and Clustering](https://mlsanity.com/paper/2301.01805/) (2023)
- [Bag of Image Patch Embedding Behind the Success of Self-Supervised Learning](https://mlsanity.com/paper/2206.08954/) (2022)
- [Unsupervised Learning of Structured Representations via Closed-Loop Transcription](https://mlsanity.com/paper/2210.16782/) (2022)
- [Exploring a Principled Framework for Deep Subspace Clustering](https://mlsanity.com/paper/2503.17288/) (2025)
- [Minimalistic Unsupervised Learning with the Sparse Manifold Transform](https://mlsanity.com/paper/2209.15261/) (2022)
- [GEDI: GEnerative and DIscriminative Training for Self-Supervised Learning](https://mlsanity.com/paper/2212.13425/) (2022)
- [Multi-objective Deep Learning: Taxonomy and Survey of the State of the Art](https://mlsanity.com/paper/2412.01566/) (2024)
- [Structuring Sparsity: Block-Sparse Featurizers Capture Visual Concept Manifolds](https://mlsanity.com/paper/2606.25234/) (2026)
- [Learning Globally Smooth Functions on Manifolds](https://mlsanity.com/paper/2210.00301/) (2022)
- [Feature Normalization Prevents Collapse of Non-contrastive Learning Dynamics](https://mlsanity.com/paper/2309.16109/) (2023)
- [Structural Adversarial Objectives for Self-Supervised Representation Learning](https://mlsanity.com/paper/2310.00357/) (2023)
- [Unifying Self-Supervised Clustering and Energy-Based Models](https://mlsanity.com/paper/2401.00873/) (2023)
- [Image Hashing via Cross-View Code Alignment in the Age of Foundation Models](https://mlsanity.com/paper/2510.27584/) (2025)
- [Self-Supervised Learning as Discrete Communication](https://mlsanity.com/paper/2602.09764/) (2026)
- [Compressing Sentence Representation with maximum Coding Rate Reduction](https://mlsanity.com/paper/2304.12674/) (2023)
- [Generating Spatial Synthetic Populations Using Wasserstein Generative Adversarial Network: A Case Study with EU-SILC Data for Helsinki and Thessaloniki](https://mlsanity.com/paper/2501.16080/) (2025)

## Abstract (arXiv)

Given a union of non-linear manifolds, non-linear subspace clustering or manifold clustering aims to cluster data points based on manifold structures and also learn to parameterize each manifold as a linear subspace in a feature space. Deep neural networks have the potential to achieve this goal under highly non-linear settings given their large capacity and flexibility. We argue that achieving manifold clustering with neural networks requires two essential ingredients: a domain-specific constraint that ensures the identification of the manifolds, and a learning algorithm for embedding each manifold to a linear subspace in the feature space. This work shows that many constraints can be implemented by data augmentation. For subspace feature learning, Maximum Coding Rate Reduction (MCR2) objective can be used. Putting them together yields em Neural Manifold Clustering and Embedding (NMCE), a novel method for general purpose manifold clustering, which significantly outperforms autoencoder-based deep subspace clustering. Further, on more challenging natural image datasets, NMCE can also outperform other algorithms specifically designed for clustering. Qualitatively, we demonstrate that NMCE learns a meaningful and interpretable feature space. As the formulation of NMCE is closely related to several important Self-supervised learning (SSL) methods, we believe this work can help us build a deeper understanding on SSL representation learning.