# End-to-end Differentiable Clustering with Associative Memories

arXiv: https://arxiv.org/abs/2306.03209 · HTML: https://mlsanity.com/paper/2306.03209/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 75 references that are in the map.
- First posted on arXiv 5 Jun 2023.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Towards K-means-friendly Spaces: Simultaneous Deep Learning and Clustering](https://mlsanity.com/paper/1610.04794/) (2016)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Deep Subspace Clustering Networks](https://mlsanity.com/paper/1709.02508/) (2017)
- [Deep Clustering: A Comprehensive Survey](https://mlsanity.com/paper/2210.04142/) (2022)
- [A Comprehensive Survey on Deep Clustering: Taxonomy, Challenges, and Future Directions](https://mlsanity.com/paper/2206.07579/) (2022)
- [Universal Hopfield Networks: A General Framework for Single-Shot Associative Memory Models](https://mlsanity.com/paper/2202.04557/) (2022)
- [Hierarchical Associative Memory](https://mlsanity.com/paper/2107.06446/) (2021)
- [Self-Supervised Convolutional Subspace Clustering Network](https://mlsanity.com/paper/1905.00149/) (2019)
- [Simplicial Hopfield networks](https://mlsanity.com/paper/2305.05179/) (2023)
- [Deep Clustering Based on a Mixture of Autoencoders](https://mlsanity.com/paper/1812.06535/) (2018)
- [Tree-Independent Dual-Tree Algorithms](https://mlsanity.com/paper/1304.4327/) (2013)
- [Automated Clustering of High-dimensional Data with a Feature Weighted Mean Shift Algorithm](https://mlsanity.com/paper/2012.10929/) (2020)
- [Plug-and-play dual-tree algorithm runtime analysis](https://mlsanity.com/paper/1501.05222/) (2015)

## Built on by (most cited first, 15 of 15)

- [Provably Optimal Memory Capacity for Modern Hopfield Models: Transformer-Compatible Dense Associative Memories as Spherical Codes](https://mlsanity.com/paper/2410.23126/) (2024)
- [Beyond Scaling Laws: Understanding Transformer Performance with Associative Memory](https://mlsanity.com/paper/2405.08707/) (2024)
- [Dense Associative Memory Through the Lens of Random Features](https://mlsanity.com/paper/2410.24153/) (2024)
- [Bridging Associative Memory and Probabilistic Modeling](https://mlsanity.com/paper/2402.10202/) (2024)
- [Differentiable Cluster Graph Neural Network](https://mlsanity.com/paper/2405.16185/) (2024)
- [Deep Clustering with Associative Memories](https://mlsanity.com/paper/2601.00963/) (2026)
- [Adaptive Hopfield Network: Rethinking Similarities in Associative Memory](https://mlsanity.com/paper/2511.20609/) (2025)
- [On the Role of Hidden States of Modern Hopfield Network in Transformer](https://mlsanity.com/paper/2511.20698/) (2025)
- [DDCL: Deep Dual Competitive Learning: A Differentiable End-to-End Framework for Unsupervised Prototype-Based Representation Learning](https://mlsanity.com/paper/2604.01740/) (2026)
- [Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization](https://mlsanity.com/paper/2604.05042/) (2026)
- [Binding threshold units with artificial oscillatory neurons](https://mlsanity.com/paper/2505.03648/) (2025)
- [Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics](https://mlsanity.com/paper/2603.20656/) (2026)
- [A Data-Informed Variational Clustering Framework for Noisy High-Dimensional Data](https://mlsanity.com/paper/2604.06864/) (2026)
- [Do Vision Transformers Need All-to-All Attention? Global Communication Through Elastic Learned Cores](https://mlsanity.com/paper/2605.12491/) (2026)
- [Controlled Dynamics Attractor Transformer](https://mlsanity.com/paper/2606.15207/) (2026)

## Abstract (arXiv)

Clustering is a widely used unsupervised learning technique involving an intensive discrete optimization problem. Associative Memory models or AMs are differentiable neural networks defining a recursive dynamical system, which have been integrated with various deep learning architectures. We uncover a novel connection between the AM dynamics and the inherent discrete assignment necessary in clustering to propose a novel unconstrained continuous relaxation of the discrete clustering problem, enabling end-to-end differentiable clustering with AM, dubbed ClAM. Leveraging the pattern completion ability of AMs, we further develop a novel self-supervised clustering loss. Our evaluations on varied datasets demonstrate that ClAM benefits from the self-supervision, and significantly improves upon both the traditional Lloyd's k-means algorithm, and more recent continuous clustering relaxations (by upto 60% in terms of the Silhouette Coefficient).