GRIL: A $2$-parameter Persistence Based Vectorization for Machine Learning

Cheng Xin, Soham Mukherjee, Shreyas N. Samaga, Tamal K. Dey
Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), PMLR 221:313-333, 2023.

Abstract

$1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for $2$-parameter persistence modules. We show that this vector representation is $1$-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of $1$-parameter and $2$-parameter persistence modules. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at [https://github.com/soham0209/mpml-graph](https://github.com/soham0209/mpml-graph).

Cite this Paper

BibTeX
@InProceedings{pmlr-v221-xin23a, title = {GRIL: A $2$-parameter Persistence Based Vectorization for Machine Learning}, author = {Xin, Cheng and Mukherjee, Soham and Samaga, Shreyas N. and Dey, Tamal K.}, booktitle = {Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)}, pages = {313--333}, year = {2023}, editor = {Doster, Timothy and Emerson, Tegan and Kvinge, Henry and Miolane, Nina and Papillon, Mathilde and Rieck, Bastian and Sanborn, Sophia}, volume = {221}, series = {Proceedings of Machine Learning Research}, month = {28 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v221/xin23a/xin23a.pdf}, url = {https://proceedings.mlr.press/v221/xin23a.html}, abstract = {$1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for $2$-parameter persistence modules. We show that this vector representation is $1$-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of $1$-parameter and $2$-parameter persistence modules. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at [https://github.com/soham0209/mpml-graph](https://github.com/soham0209/mpml-graph).} }
Endnote
%0 Conference Paper %T GRIL: A $2$-parameter Persistence Based Vectorization for Machine Learning %A Cheng Xin %A Soham Mukherjee %A Shreyas N. Samaga %A Tamal K. Dey %B Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) %C Proceedings of Machine Learning Research %D 2023 %E Timothy Doster %E Tegan Emerson %E Henry Kvinge %E Nina Miolane %E Mathilde Papillon %E Bastian Rieck %E Sophia Sanborn %F pmlr-v221-xin23a %I PMLR %P 313--333 %U https://proceedings.mlr.press/v221/xin23a.html %V 221 %X $1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for $2$-parameter persistence modules. We show that this vector representation is $1$-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of $1$-parameter and $2$-parameter persistence modules. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at [https://github.com/soham0209/mpml-graph](https://github.com/soham0209/mpml-graph).
APA
Xin, C., Mukherjee, S., Samaga, S.N. & Dey, T.K.. (2023). GRIL: A $2$-parameter Persistence Based Vectorization for Machine Learning. Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), in Proceedings of Machine Learning Research 221:313-333 Available from https://proceedings.mlr.press/v221/xin23a.html.

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