# Persistence Spheres: Bi-continuous Representations of Persistence Diagrams

arXiv: https://arxiv.org/abs/2509.16999 · HTML: https://mlsanity.com/paper/2509.16999/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 40 references that are in the map.
- First posted on arXiv 21 Sep 2025.

## Builds on (most cited first, 12 of 12)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Kernel method for persistence diagrams via kernel embedding and weight factor](https://mlsanity.com/paper/1706.03472/) (2017)
- [Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport](https://mlsanity.com/paper/1901.03048/) (2019)
- [A persistence landscapes toolbox for topological statistics](https://mlsanity.com/paper/1501.00179/) (2014)
- [scikit-fda: A Python Package for Functional Data Analysis](https://mlsanity.com/paper/2211.02566/) (2022)
- [Comparing persistence diagrams through complex vectors](https://mlsanity.com/paper/1505.01335/) (2015)
- [On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces](https://mlsanity.com/paper/1806.06924/) (2018)
- [Persistent Homology for Breast Tumor Classification using Mammogram Scans](https://mlsanity.com/paper/2201.02295/) (2022)
- [Topological data analysis for random sets and its application in detecting outliers and goodness of fit testing](https://mlsanity.com/paper/2404.04702/) (2024)

## Abstract (arXiv)

We introduce persistence spheres, a novel functional representation of persistence diagrams. Unlike existing embeddings (such as persistence images, landscapes, or kernel methods), persistence spheres provide a bi-continuous mapping: they are Lipschitz continuous with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This ensures, in a theoretically optimal way, both stability and geometric fidelity, making persistence spheres the representation that most closely mirrors the Wasserstein geometry of PDs in linear space. We derive explicit formulas for persistence spheres, showing that they can be computed efficiently and parallelized with minimal overhead. Empirically, we evaluate them on diverse regression and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres consistently deliver state-of-the-art or competitive performance compared to persistence images, persistence landscapes, and the sliced Wasserstein kernel.