# Stability for Inference with Persistent Homology Rank Functions

arXiv: https://arxiv.org/abs/2307.02904 · HTML: https://mlsanity.com/paper/2307.02904/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 91 references that are in the map.
- First posted on arXiv 6 Jul 2023.

## Builds on (most cited first, 16 of 16)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Stochastic Convergence of Persistence Landscapes and Silhouettes](https://mlsanity.com/paper/1312.0308/) (2013)
- [Interactive Visualization of 2-D Persistence Modules](https://mlsanity.com/paper/1512.00180/) (2015)
- [Confidence sets for persistence diagrams](https://mlsanity.com/paper/1303.7117/) (2013)
- [A Kernel for Multi-Parameter Persistent Homology](https://mlsanity.com/paper/1809.10231/) (2018)
- [Generalized Persistence Diagrams for Persistence Modules over Posets](https://mlsanity.com/paper/1810.11517/) (2018)
- [Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions](https://mlsanity.com/paper/2107.06800/) (2021)
- [The rank invariant stability via interleavings](https://mlsanity.com/paper/1412.3374/) (2014)
- [Metrics for generalized persistence modules](https://mlsanity.com/paper/1312.3829/) (2013)
- [Computing multiparameter persistent homology through a discrete Morse-based approach](https://mlsanity.com/paper/1811.05396/) (2018)
- [Exact computation of the matching distance on 2-parameter persistence modules](https://mlsanity.com/paper/1812.09085/) (2018)
- [Efficient Approximation of the Matching Distance for 2-parameter persistence](https://mlsanity.com/paper/1912.05826/) (2019)
- [A Geometric Condition for Uniqueness of Fr'echet Means of Persistence Diagrams](https://mlsanity.com/paper/2207.03943/) (2022)
- [k-Means Clustering for Persistent Homology](https://mlsanity.com/paper/2210.10003/) (2022)

## Built on by (most cited first, 1 of 1)

- [Is magnitude 'generically continuous' for finite metric spaces?](https://mlsanity.com/paper/2501.08745/) (2025)

## Abstract (arXiv)

Persistent homology barcodes and diagrams are a cornerstone of topological data analysis that capture the "shape" of a wide range of complex data structures, such as point clouds, networks, and functions. However, their use in statistical settings is challenging due to their complex geometric structure. In this paper, we revisit the persistent homology rank function, which is mathematically equivalent to a barcode and persistence diagram, as a tool for statistics and machine learning. Rank functions, being functions, enable the direct application of the statistical theory of functional data analysis (FDA)-a domain of statistics adapted for data in the form of functions. A key challenge they present over barcodes in practice, however, is their lack of stability-a property that is crucial to validate their use as a faithful representation of the data and therefore a viable summary statistic. In this paper, we fill this gap by deriving two stability results for persistent homology rank functions under a suitable metric for FDA integration. We then study the performance of rank functions in functional inferential statistics and machine learning on real data applications, in both single and multiparameter persistent homology. We find that the use of persistent homology captured by rank functions offers a clear improvement over existing non-persistence-based approaches.