# Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams

arXiv: https://arxiv.org/abs/1802.03569 · HTML: https://mlsanity.com/paper/1802.03569/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2018.

## Built on by (most cited first, 30 of 33)

- [Topological Relational Learning on Graphs](https://mlsanity.com/paper/2110.15529/) (2021)
- [RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds](https://mlsanity.com/paper/2202.01725/) (2022)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [Entropy Partial Transport with Tree Metrics: Theory and Practice](https://mlsanity.com/paper/2101.09756/) (2021)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications](https://mlsanity.com/paper/2311.05116/) (2023)
- [Graphcode: Learning from multiparameter persistent homology using graph neural networks](https://mlsanity.com/paper/2405.14302/) (2024)
- [Learning Hyperbolic Representations of Topological Features](https://mlsanity.com/paper/2103.09273/) (2021)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [Augmenting Molecular Deep Generative Models with Topological Data Analysis Representations](https://mlsanity.com/paper/2106.04464/) (2021)
- [A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.02836/) (2026)
- [Statistical learning on measures: an application to persistence diagrams](https://mlsanity.com/paper/2303.08456/) (2023)
- [A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.04046/) (2026)
- [A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering](https://mlsanity.com/paper/2105.12208/) (2021)
- [Smart Vectorizations for Single and Multiparameter Persistence](https://mlsanity.com/paper/2104.04787/) (2021)
- [Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology](https://mlsanity.com/paper/2109.10960/) (2021)
- [Consistent Interpolating Ensembles via the Manifold-Hilbert Kernel](https://mlsanity.com/paper/2205.09342/) (2022)
- [Discrete transforms of quantized persistence diagrams](https://mlsanity.com/paper/2312.17093/) (2023)
- [Scalable Sobolev IPM for Probability Measures on a Graph](https://mlsanity.com/paper/2502.00737/) (2025)
- [From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features](https://mlsanity.com/paper/2606.11911/) (2026)
- [A Mathematical Framework for Topological Causal Data Analysis](https://mlsanity.com/paper/2607.28161/) (2026)
- [A Multi-scale Graph Signature for Persistence Diagrams based on Return Probabilities of Random Walks](https://mlsanity.com/paper/2209.14264/) (2022)
- [Metric Distribution to Vector: Constructing Data Representation via Broad-Scale Discrepancies](https://mlsanity.com/paper/2210.00415/) (2022)
- [An Evolution Kernel Method for Graph Classification through Heat Diffusion Dynamics](https://mlsanity.com/paper/2306.14688/) (2023)
- [Persistence kernels for classification: A comparative study](https://mlsanity.com/paper/2408.07090/) (2024)
- [Is Smoothness the Key to Robustness? A Comparison of Attention and Convolution Models Using a Novel Metric](https://mlsanity.com/paper/2410.17628/) (2024)
- [FILTR: Extracting Topological Features from Pretrained 3D Models](https://mlsanity.com/paper/2604.22334/) (2026)

## Abstract (arXiv)

Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data such as shapes, linked twist maps, and material data. Among them, textitpersistent homology is a well-known tool to extract robust topological features, and outputs as textitpersistence diagrams (PDs). However, PDs are point multi-sets which can not be used in machine learning algorithms for vector data. To deal with it, an emerged approach is to use kernel methods, and an appropriate geometry for PDs is an important factor to measure the similarity of PDs. A popular geometry for PDs is the textitWasserstein metric. However, Wasserstein distance is not textitnegative definite. Thus, it is limited to build positive definite kernels upon the Wasserstein distance textitwithout approximation. In this work, we rely upon the alternative textitFisher information geometry to propose a positive definite kernel for PDs textitwithout approximation, namely the Persistence Fisher (PF) kernel. Then, we analyze eigensystem of the integral operator induced by the proposed kernel for kernel machines. Based on that, we derive generalization error bounds via covering numbers and Rademacher averages for kernel machines with the PF kernel. Additionally, we show some nice properties such as stability and infinite divisibility for the proposed kernel. Furthermore, we also propose a linear time complexity over the number of points in PDs for an approximation of our proposed kernel with a bounded error. Throughout experiments with many different tasks on various benchmark datasets, we illustrate that the PF kernel compares favorably with other baseline kernels for PDs.