GRIL: A 2-Parameter Persistence Based Vectorization for Machine Learning

Abstract

One-parameter persistent homology has become a standard tool for extracting topological features from data, but richer data descriptions often require multiple filtration parameters. This paper introduces the Generalized Rank Invariant Landscape (GRIL), a vector representation for 2-parameter persistence modules designed for machine learning. GRIL is stable, differentiable with respect to underlying filtration functions, and efficiently computable. We evaluate the representation on synthetic and graph-learning benchmarks, including experiments that augment graph neural networks with GRIL features.

Presentation

This paper appeared in the ICML 2023 TAG-ML workshop proceedings and was selected for an oral presentation.

Key Contributions

• 2-parameter vectorization: Introduces GRIL as a machine-learning-ready vector representation for 2-parameter persistence modules.
• Stability and differentiability: Proves properties needed to use GRIL robustly in learning pipelines.
• Efficient computation: Provides an algorithmic approach for computing the representation.
• Graph learning applications: Tests GRIL on graph benchmarks and uses it to augment graph neural networks.

Citation

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

BibTeX Citation

@inproceedings{xin2023gril,
  title={GRIL: A 2-Parameter Persistence Based Vectorization for Machine Learning},
  author={Cheng Xin and Soham Mukherjee and Shreyas N. Samaga and Tamal K. Dey},
  booktitle={Proceedings of the 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)},
  series={Proceedings of Machine Learning Research},
  volume={221},
  pages={313--333},
  year={2023},
  url={https://proceedings.mlr.press/v221/xin23a.html}
}

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