# Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH

arXiv: https://arxiv.org/abs/2310.16437 · HTML: https://mlsanity.com/paper/2310.16437/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 25 Oct 2023.

## Builds on (most cited first, 9 of 9)

- [Neural Architecture Search with Bayesian Optimisation and Optimal Transport](https://mlsanity.com/paper/1802.07191/) (2018)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Wasserstein Embedding for Graph Learning](https://mlsanity.com/paper/2006.09430/) (2020)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Connectivity-Optimized Representation Learning via Persistent Homology](https://mlsanity.com/paper/1906.09003/) (2019)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Fast Optimal Transport Averaging of Neuroimaging Data](https://mlsanity.com/paper/1503.08596/) (2015)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)

## Built on by (most cited first, 3 of 3)

- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [A Persistent Homology Design Space for 3D Point Cloud Deep Learning](https://mlsanity.com/paper/2604.04299/) (2026)
- [Prototype Selection Using Topological Data Analysis](https://mlsanity.com/paper/2511.04873/) (2025)

## Abstract (arXiv)

Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration. Most filtrations used for persistent homology depend (implicitly) on a chosen metric, which is typically agnostically chosen as the standard Euclidean metric on ℝn. Recent work has tried to uncover the 'true' metric on the point cloud using distance-to-measure functions, in order to obtain more meaningful persistent homology results. Here we propose an alternative look at this problem: we posit that information on the point cloud is lost when restricting persistent homology to a single (correct) distance function. Instead, we show how by varying the distance function on the underlying space and analysing the corresponding shifts in the persistence diagrams, we can extract additional topological and geometrical information. Finally, we numerically show that non-isotropic persistent homology can extract information on orientation, orientational variance, and scaling of randomly generated point clouds with good accuracy and conduct some experiments on real-world data.