# A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams

arXiv: https://arxiv.org/abs/2402.14489 · HTML: https://mlsanity.com/paper/2402.14489/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 71 references that are in the map.
- First posted on arXiv 22 Feb 2024.

## Builds on (most cited first, 25 of 25)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Geometry Score: A Method For Comparing Generative Adversarial Networks](https://mlsanity.com/paper/1802.02664/) (2018)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Persistence Curves: A canonical framework for summarizing persistence diagrams](https://mlsanity.com/paper/1904.07768/) (2019)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [Scalable Nearest Neighbor Search for Optimal Transport](https://mlsanity.com/paper/1910.04126/) (2019)
- [Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning](https://mlsanity.com/paper/1812.11832/) (2018)
- [A Topological "Reading" Lesson: Classification of MNIST using TDA](https://mlsanity.com/paper/1910.08345/) (2019)
- [Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling](https://mlsanity.com/paper/1801.02790/) (2018)
- [The importance of the whole: topological data analysis for the network neuroscientist](https://mlsanity.com/paper/1806.05167/) (2018)
- [Persistent Homology of Attractors For Action Recognition](https://mlsanity.com/paper/1603.05310/) (2016)
- [Topology Distance: A Topology-Based Approach For Evaluating Generative Adversarial Networks](https://mlsanity.com/paper/2002.12054/) (2020)
- [Approximation algorithms for 1-Wasserstein distance between persistence diagrams](https://mlsanity.com/paper/2104.07710/) (2021)
- [Shape retrieval of non-rigid 3d human models](https://mlsanity.com/paper/2003.08763/) (2020)
- [Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification](https://mlsanity.com/paper/2110.14734/) (2021)
- [Universality of persistence diagrams and the bottleneck and Wasserstein distances](https://mlsanity.com/paper/1912.02563/) (2019)

## Abstract (arXiv)

Persistence diagrams (PD)s play a central role in topological data analysis, and are used in an ever increasing variety of applications. The comparison of PD data requires computing comparison metrics among large sets of PDs, with metrics which are accurate, theoretically sound, and fast to compute. Especially for denser multi-dimensional PDs, such comparison metrics are lacking. While on the one hand, Wasserstein-type distances have high accuracy and theoretical guarantees, they incur high computational cost. On the other hand, distances between vectorizations such as Persistence Statistics (PS)s have lower computational cost, but lack the accuracy guarantees and in general they are not guaranteed to distinguish PDs (i.e. the two PS vectors of different PDs may be equal). In this work we introduce a class of pseudodistances called Extended Topological Pseudodistances (ETD)s, which have tunable complexity, and can approximate Sliced and classical Wasserstein distances at the high-complexity extreme, while being computationally lighter and close to Persistence Statistics at the lower complexity extreme, and thus allow users to interpolate between the two metrics. We build theoretical comparisons to show how to fit our new distances at an intermediate level between persistence vectorizations and Wasserstein distances. We also experimentally verify that ETDs outperform PSs in terms of accuracy and outperform Wasserstein and Sliced Wasserstein distances in terms of computational complexity.