The Persistent Laplacian for Data Science: Evaluating Higher-Order Persistent Spectral Representations of Data

Thomas Davies, Zhengchao Wan, Ruben J Sanchez-Garcia
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:7249-7263, 2023.

Abstract

Persistent homology is arguably the most successful technique in Topological Data Analysis. It combines homology, a topological feature of a data set, with persistence, which tracks the evolution of homology over different scales. The persistent Laplacian is a recent theoretical development that combines persistence with the combinatorial Laplacian, the higher-order extension of the well-known graph Laplacian. Crucially, the Laplacian encode both the homology of a data set, and some additional geometric information not captured by the homology. Here, we provide the first investigation into the efficacy of the persistence Laplacian as an embedding of data for downstream classification and regression tasks. We extend the persistent Laplacian to cubical complexes so it can be used on images, then evaluate its performance as an embedding method on the MNIST and MoleculeNet datasets, demonstrating that it consistently outperforms persistent homology across tasks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-davies23c, title = {The Persistent {L}aplacian for Data Science: Evaluating Higher-Order Persistent Spectral Representations of Data}, author = {Davies, Thomas and Wan, Zhengchao and Sanchez-Garcia, Ruben J}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {7249--7263}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/davies23c/davies23c.pdf}, url = {https://proceedings.mlr.press/v202/davies23c.html}, abstract = {Persistent homology is arguably the most successful technique in Topological Data Analysis. It combines homology, a topological feature of a data set, with persistence, which tracks the evolution of homology over different scales. The persistent Laplacian is a recent theoretical development that combines persistence with the combinatorial Laplacian, the higher-order extension of the well-known graph Laplacian. Crucially, the Laplacian encode both the homology of a data set, and some additional geometric information not captured by the homology. Here, we provide the first investigation into the efficacy of the persistence Laplacian as an embedding of data for downstream classification and regression tasks. We extend the persistent Laplacian to cubical complexes so it can be used on images, then evaluate its performance as an embedding method on the MNIST and MoleculeNet datasets, demonstrating that it consistently outperforms persistent homology across tasks.} }
Endnote
%0 Conference Paper %T The Persistent Laplacian for Data Science: Evaluating Higher-Order Persistent Spectral Representations of Data %A Thomas Davies %A Zhengchao Wan %A Ruben J Sanchez-Garcia %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-davies23c %I PMLR %P 7249--7263 %U https://proceedings.mlr.press/v202/davies23c.html %V 202 %X Persistent homology is arguably the most successful technique in Topological Data Analysis. It combines homology, a topological feature of a data set, with persistence, which tracks the evolution of homology over different scales. The persistent Laplacian is a recent theoretical development that combines persistence with the combinatorial Laplacian, the higher-order extension of the well-known graph Laplacian. Crucially, the Laplacian encode both the homology of a data set, and some additional geometric information not captured by the homology. Here, we provide the first investigation into the efficacy of the persistence Laplacian as an embedding of data for downstream classification and regression tasks. We extend the persistent Laplacian to cubical complexes so it can be used on images, then evaluate its performance as an embedding method on the MNIST and MoleculeNet datasets, demonstrating that it consistently outperforms persistent homology across tasks.
APA
Davies, T., Wan, Z. & Sanchez-Garcia, R.J.. (2023). The Persistent Laplacian for Data Science: Evaluating Higher-Order Persistent Spectral Representations of Data. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:7249-7263 Available from https://proceedings.mlr.press/v202/davies23c.html.

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