# Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport

arXiv: https://arxiv.org/abs/1805.08331 · HTML: https://mlsanity.com/paper/1805.08331/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2018.

## Built on by (most cited first, 30 of 32)

- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [TopoCellGen: Generating Histopathology Cell Topology with a Diffusion Model](https://mlsanity.com/paper/2412.06011/) (2024)
- [Entropy Partial Transport with Tree Metrics: Theory and Practice](https://mlsanity.com/paper/2101.09756/) (2021)
- [Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study](https://mlsanity.com/paper/2207.14080/) (2022)
- [Semi-supervised Segmentation of Histopathology Images with Noise-Aware Topological Consistency](https://mlsanity.com/paper/2311.16447/) (2023)
- [Estimation and Quantization of Expected Persistence Diagrams](https://mlsanity.com/paper/2105.04852/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [Learning to Generate Wasserstein Barycenters](https://mlsanity.com/paper/2102.12178/) (2021)
- [Fast Topological Clustering with Wasserstein Distance](https://mlsanity.com/paper/2112.00101/) (2021)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [Wasserstein Dictionaries of Persistence Diagrams](https://mlsanity.com/paper/2304.14852/) (2023)
- [Optimal Transport on the Lie Group of Roto-translations](https://mlsanity.com/paper/2402.15322/) (2024)
- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measures](https://mlsanity.com/paper/2104.01672/) (2021)
- [A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering](https://mlsanity.com/paper/2105.12208/) (2021)
- [Local Topology Measures of Contextual Language Model Latent Spaces With Applications to Dialogue Term Extraction](https://mlsanity.com/paper/2408.03706/) (2024)
- [An Homogeneous Unbalanced Regularized Optimal Transport model with applications to Optimal Transport with Boundary](https://mlsanity.com/paper/2201.02082/) (2022)
- [k-Means Clustering for Persistent Homology](https://mlsanity.com/paper/2210.10003/) (2022)
- [Reliable Malware Analysis and Detection using Topology Data Analysis](https://mlsanity.com/paper/2211.01535/) (2022)
- [Identifying visible tissue in intraoperative ultrasound: a method and application](https://mlsanity.com/paper/2306.01190/) (2023)
- [Towards Scalable Topological Regularizers](https://mlsanity.com/paper/2501.14641/) (2025)
- [Robust Barycenters of Persistence Diagrams](https://mlsanity.com/paper/2509.14904/) (2025)
- [Topological Classification in a Wasserstein Distance Based Vector Space](https://mlsanity.com/paper/2202.01275/) (2022)
- [Convolutional Persistence Transforms](https://mlsanity.com/paper/2208.02107/) (2022)
- [On topological data analysis for structural dynamics: an introduction to persistent homology](https://mlsanity.com/paper/2209.05134/) (2022)
- [A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams](https://mlsanity.com/paper/2402.14489/) (2024)
- [Topology Aware Neural Interpolation of Scalar Fields](https://mlsanity.com/paper/2508.17995/) (2025)
- [Continuous Edit Distance, Geodesics and Barycenters of Time-varying Persistence Diagrams](https://mlsanity.com/paper/2512.12939/) (2025)
- [From Classical to Topological Neural Networks Under Uncertainty](https://mlsanity.com/paper/2602.10266/) (2026)

## Abstract (arXiv)

Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not Hilbertian. Indeed, this was recently exemplified in a string of papers which show that the simple task of averaging a few PDs can be computationally prohibitive. We propose in this article a tractable framework to carry out standard tasks on PDs at scale, notably evaluating distances, estimating barycenters and performing clustering. This framework builds upon a reformulation of PD metrics as optimal transport (OT) problems. Doing so, we can exploit recent computational advances: the OT problem on a planar grid, when regularized with entropy, is convex can be solved in linear time using the Sinkhorn algorithm and convolutions. This results in scalable computations that can stream on GPUs. We demonstrate the efficiency of our approach by carrying out clustering with diagrams metrics on several thousands of PDs, a scale never seen before in the literature.