PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures

Mathieu Carriere, Frederic Chazal, Yuichi Ike, Theo Lacombe, Martin Royer, Yuhei Umeda
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:2786-2796, 2020.

Abstract

Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the metric space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Machine Learning techniques. To address this concern, several vectorization methods have been put forward that embed persistence diagrams into either finite-dimensional Euclidean space or implicit infinite dimensional Hilbert space with kernels. In this work, we focus on persistence diagrams built on top of graphs. Relying on extended persistence theory and the so-called heat kernel signature, we show how graphs can be encoded by (extended) persistence diagrams in a provably stable way. We then propose a general and versatile framework for learning vectorizations of persistence diagrams, which encompasses most of the vectorization techniques used in the literature. We finally showcase the experimental strength of our setup by achieving competitive scores on classification tasks on real-life graph datasets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v108-carriere20a, title = {PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures}, author = {Carriere, Mathieu and Chazal, Frederic and Ike, Yuichi and Lacombe, Theo and Royer, Martin and Umeda, Yuhei}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {2786--2796}, year = {2020}, editor = {Chiappa, Silvia and Calandra, Roberto}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/carriere20a/carriere20a.pdf}, url = {https://proceedings.mlr.press/v108/carriere20a.html}, abstract = {Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the metric space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Machine Learning techniques. To address this concern, several vectorization methods have been put forward that embed persistence diagrams into either finite-dimensional Euclidean space or implicit infinite dimensional Hilbert space with kernels. In this work, we focus on persistence diagrams built on top of graphs. Relying on extended persistence theory and the so-called heat kernel signature, we show how graphs can be encoded by (extended) persistence diagrams in a provably stable way. We then propose a general and versatile framework for learning vectorizations of persistence diagrams, which encompasses most of the vectorization techniques used in the literature. We finally showcase the experimental strength of our setup by achieving competitive scores on classification tasks on real-life graph datasets.} }
Endnote
%0 Conference Paper %T PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures %A Mathieu Carriere %A Frederic Chazal %A Yuichi Ike %A Theo Lacombe %A Martin Royer %A Yuhei Umeda %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-carriere20a %I PMLR %P 2786--2796 %U https://proceedings.mlr.press/v108/carriere20a.html %V 108 %X Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the metric space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Machine Learning techniques. To address this concern, several vectorization methods have been put forward that embed persistence diagrams into either finite-dimensional Euclidean space or implicit infinite dimensional Hilbert space with kernels. In this work, we focus on persistence diagrams built on top of graphs. Relying on extended persistence theory and the so-called heat kernel signature, we show how graphs can be encoded by (extended) persistence diagrams in a provably stable way. We then propose a general and versatile framework for learning vectorizations of persistence diagrams, which encompasses most of the vectorization techniques used in the literature. We finally showcase the experimental strength of our setup by achieving competitive scores on classification tasks on real-life graph datasets.
APA
Carriere, M., Chazal, F., Ike, Y., Lacombe, T., Royer, M. & Umeda, Y.. (2020). PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:2786-2796 Available from https://proceedings.mlr.press/v108/carriere20a.html.

Related Material