# Wasserstein Dictionaries of Persistence Diagrams

arXiv: https://arxiv.org/abs/2304.14852 · HTML: https://mlsanity.com/paper/2304.14852/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 108 references that are in the map.
- First posted on arXiv 28 Apr 2023.

## Builds on (most cited first, 19 of 19)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning](https://mlsanity.com/paper/1708.01955/) (2017)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Task-based Augmented Contour Trees with Fibonacci Heaps](https://mlsanity.com/paper/1902.04805/) (2019)
- [Progressive Wasserstein Barycenters of Persistence Diagrams](https://mlsanity.com/paper/1907.04565/) (2019)
- [Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-aware Matchings](https://mlsanity.com/paper/1908.07841/) (2019)
- [Persistence Atlas for Critical Point Variability in Ensembles](https://mlsanity.com/paper/1807.11212/) (2018)
- [Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data -- An Algorithm and A Benchmark](https://mlsanity.com/paper/2206.13932/) (2022)
- [Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study](https://mlsanity.com/paper/2207.14080/) (2022)
- [Uncertainty Visualization of 2D Morse Complex Ensembles Using Statistical Summary Maps](https://mlsanity.com/paper/1912.06341/) (2019)
- [Topological data analysis of vortices in the magnetically-induced current density in LiH molecule](https://mlsanity.com/paper/2212.08690/) (2022)
- [A Structural Average of Labeled Merge Trees for Uncertainty Visualization](https://mlsanity.com/paper/1908.00113/) (2019)
- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [Utilising urgent computing to tackle the spread of mosquito-borne diseases](https://mlsanity.com/paper/2111.05654/) (2021)

## Built on by (most cited first, 4 of 4)

- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [TTK is Getting MPI-Ready](https://mlsanity.com/paper/2310.08339/) (2023)
- [Robust Barycenters of Persistence Diagrams](https://mlsanity.com/paper/2509.14904/) (2025)
- [Sierpi'nski--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation](https://mlsanity.com/paper/2609.01528/) (2026)

## Abstract (arXiv)

This paper presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters [100], [102] of a dictionary of atom diagrams. We introduce a multi-scale gradient descent approach for the efficient resolution of the corresponding minimization problem, which interleaves the optimization of the barycenter weights with the optimization of the atom diagrams. Our approach leverages the analytic expressions for the gradient of both sub-problems to ensure fast iterations and it additionally exploits shared-memory parallelism. Extensive experiments on public ensembles demonstrate the efficiency of our approach, with Wasserstein dictionary computations in the orders of minutes for the largest examples. We show the utility of our contributions in two applications. First, we apply Wassserstein dictionaries to data reduction and reliably compress persistence diagrams by concisely representing them with their weights in the dictionary. Second, we present a dimensionality reduction framework based on a Wasserstein dictionary defined with a small number of atoms (typically three) and encode the dictionary as a low dimensional simplex embedded in a visual space (typically in 2D). In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.