# Topological Classification in a Wasserstein Distance Based Vector Space

arXiv: https://arxiv.org/abs/2202.01275 · HTML: https://mlsanity.com/paper/2202.01275/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 54 references that are in the map.
- First posted on arXiv 2 Feb 2022.

## Builds on (most cited first, 14 of 14)

- [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)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Propagation Kernels](https://mlsanity.com/paper/1410.3314/) (2014)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Graph Kernels: State-of-the-Art and Future Challenges](https://mlsanity.com/paper/2011.03854/) (2020)
- [GraKeL: A Graph Kernel Library in Python](https://mlsanity.com/paper/1806.02193/) (2018)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces](https://mlsanity.com/paper/1806.06924/) (2018)

## Abstract (arXiv)

Classification of large and dense networks based on topology is very difficult due to the computational challenges of extracting meaningful topological features from real-world networks. In this paper we present a computationally tractable approach to topological classification of networks by using principled theory from persistent homology and optimal transport to define a novel vector representation for topological features. The proposed vector space is based on the Wasserstein distance between persistence barcodes. The 1-skeleton of the network graph is employed to obtain 1-dimensional persistence barcodes that represent connected components and cycles. These barcodes and the corresponding Wasserstein distance can be computed very efficiently. The effectiveness of the proposed vector space is demonstrated using support vector machines to classify simulated networks and measured functional brain networks.