# Topological Federated Clustering via Gravitational Potential Fields under Local Differential Privacy

arXiv: https://arxiv.org/abs/2512.00849 · HTML: https://mlsanity.com/paper/2512.00849/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 16 references that are in the map.
- First posted on arXiv 30 Nov 2025.

## Builds on (most cited first, 6 of 6)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [An Efficient Framework for Clustered Federated Learning](https://mlsanity.com/paper/2006.04088/) (2020)
- [Clustered Federated Learning: Model-Agnostic Distributed Multi-Task Optimization under Privacy Constraints](https://mlsanity.com/paper/1910.01991/) (2019)
- [Heterogeneity for the Win: One-Shot Federated Clustering](https://mlsanity.com/paper/2103.00697/) (2021)
- [Differentially Private Federated k-Means Clustering with Server-Side Data](https://mlsanity.com/paper/2506.05408/) (2025)
- [Fair Federated Data Clustering through Personalization: Bridging the Gap between Diverse Data Distributions](https://mlsanity.com/paper/2407.04302/) (2024)

## Built on by (most cited first, 2 of 2)

- [Air-Know: Arbiter-Calibrated Knowledge-Internalizing Robust Network for Composed Image Retrieval](https://mlsanity.com/paper/2604.19386/) (2026)
- [ConeSep: Cone-based Robust Noise-Unlearning Compositional Network for Composed Image Retrieval](https://mlsanity.com/paper/2604.20358/) (2026)

## Abstract (arXiv)

Clustering non-independent and identically distributed (non-IID) data under local differential privacy (LDP) in federated settings presents a critical challenge: preserving privacy while maintaining accuracy without iterative communication. Existing one-shot methods rely on unstable pairwise centroid distances or neighborhood rankings, degrading severely under strong LDP noise and data heterogeneity. We present Gravitational Federated Clustering (GFC), a novel approach to privacy-preserving federated clustering that overcomes the limitations of distance-based methods under varying LDP. Addressing the critical challenge of clustering non-IID data with diverse privacy guarantees, GFC transforms privatized client centroids into a global gravitational potential field where true cluster centers emerge as topologically persistent singularities. Our framework introduces two key innovations: (1) a client-side compactness-aware perturbation mechanism that encodes local cluster geometry as "mass" values, and (2) a server-side topological aggregation phase that extracts stable centroids through persistent homology analysis of the potential field's superlevel sets. Theoretically, we establish a closed-form bound between the privacy budget ε and centroid estimation error, proving the potential field's Lipschitz smoothing properties exponentially suppress noise in high-density regions. Empirically, GFC outperforms state-of-the-art methods on ten benchmarks, especially under strong LDP constraints (ε < 1), while maintaining comparable performance at lower privacy budgets. By reformulating federated clustering as a topological persistence problem in a synthetic physics-inspired space, GFC achieves unprecedented privacy-accuracy trade-offs without iterative communication, providing a new perspective for privacy-preserving distributed learning.