# Learning metrics for persistence-based summaries and applications for graph classification

arXiv: https://arxiv.org/abs/1904.12189 · HTML: https://mlsanity.com/paper/1904.12189/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Apr 2019.

## Built on by (most cited first, 30 of 57)

- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [Evaluation Metrics for Graph Generative Models: Problems, Pitfalls, and Practical Solutions](https://mlsanity.com/paper/2106.01098/) (2021)
- [Link Prediction with Persistent Homology: An Interactive View](https://mlsanity.com/paper/2102.10255/) (2021)
- [Topological Detection of Trojaned Neural Networks](https://mlsanity.com/paper/2106.06469/) (2021)
- [Topological Relational Learning on Graphs](https://mlsanity.com/paper/2110.15529/) (2021)
- [Cycle Representation Learning for Inductive Relation Prediction](https://mlsanity.com/paper/2110.02510/) (2021)
- [Going beyond persistent homology using persistent homology](https://mlsanity.com/paper/2311.06152/) (2023)
- [On the Expressivity of Persistent Homology in Graph Learning](https://mlsanity.com/paper/2302.09826/) (2023)
- [Neural Approximation of Graph Topological Features](https://mlsanity.com/paper/2201.12032/) (2022)
- [Exploring the Geometry and Topology of Neural Network Loss Landscapes](https://mlsanity.com/paper/2102.00485/) (2021)
- [PyG-SSL: A Graph Self-Supervised Learning Toolkit](https://mlsanity.com/paper/2412.21151/) (2024)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [Estimation and Quantization of Expected Persistence Diagrams](https://mlsanity.com/paper/2105.04852/) (2021)
- [Cycle Invariant Positional Encoding for Graph Representation Learning](https://mlsanity.com/paper/2311.14333/) (2023)
- [Learning Hyperbolic Representations of Topological Features](https://mlsanity.com/paper/2103.09273/) (2021)
- [A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions](https://mlsanity.com/paper/2306.11170/) (2023)
- [Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms](https://mlsanity.com/paper/2410.02622/) (2024)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [Topology Applied to Machine Learning: From Global to Local](https://mlsanity.com/paper/2103.05796/) (2021)
- [Two-level Graph Neural Network](https://mlsanity.com/paper/2201.01190/) (2022)
- [Graph similarity learning for change-point detection in dynamic networks](https://mlsanity.com/paper/2203.15470/) (2022)
- [Rethinking Persistent Homology for Visual Recognition](https://mlsanity.com/paper/2207.04220/) (2022)
- [Rethinking the Effectiveness of Graph Classification Datasets in Benchmarks for Assessing GNNs](https://mlsanity.com/paper/2407.04999/) (2024)
- [A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.02836/) (2026)
- [EGC2: Enhanced Graph Classification with Easy Graph Compression](https://mlsanity.com/paper/2107.07737/) (2021)
- [Exponential Quantum Communication Advantage in Distributed Inference and Learning](https://mlsanity.com/paper/2310.07136/) (2023)
- [A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.04046/) (2026)
- [Optimisation of Spectral Wavelets for Persistence-based Graph Classification](https://mlsanity.com/paper/2101.05201/) (2021)
- [A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering](https://mlsanity.com/paper/2105.12208/) (2021)

## Abstract (arXiv)

Recently a new feature representation and data analysis methodology based on a topological tool called persistent homology (and its corresponding persistence diagram summary) has started to attract momentum. A series of methods have been developed to map a persistence diagram to a vector representation so as to facilitate the downstream use of machine learning tools, and in these approaches, the importance (weight) of different persistence features are often preset. However often in practice, the choice of the weight function should depend on the nature of the specific type of data one considers, and it is thus highly desirable to learn a best weight function (and thus metric for persistence diagrams) from labelled data. We study this problem and develop a new weighted kernel, called WKPI, for persistence summaries, as well as an optimization framework to learn a good metric for persistence summaries. Both our kernel and optimization problem have nice properties. We further apply the learned kernel to the challenging task of graph classification, and show that our WKPI-based classification framework obtains similar or (sometimes significantly) better results than the best results from a range of previous graph classification frameworks on a collection of benchmark datasets.