# From Geometry to Topology: Inverse Theorems for Distributed Persistence

arXiv: https://arxiv.org/abs/2101.12288 · HTML: https://mlsanity.com/paper/2101.12288/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 41 references that are in the map.
- First posted on arXiv 28 Jan 2021.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [The persistence landscape and some of its properties](https://mlsanity.com/paper/1810.04963/) (2018)
- [Subsampling Methods for Persistent Homology](https://mlsanity.com/paper/1406.1901/) (2014)
- [Embeddings of Persistence Diagrams into Hilbert Spaces](https://mlsanity.com/paper/1905.05604/) (2019)
- [On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces](https://mlsanity.com/paper/1806.06924/) (2018)
- [Inverse Problems in Topological Persistence](https://mlsanity.com/paper/1810.10813/) (2018)
- [Nonembeddability of Persistence Diagrams with p>2 Wasserstein Metric](https://mlsanity.com/paper/1910.13935/) (2019)

## Built on by (most cited first, 10 of 10)

- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Spherical Rotation Dimension Reduction with Geometric Loss Functions](https://mlsanity.com/paper/2204.10975/) (2022)
- [Nonparametric Multi-shape Modeling with Uncertainty Quantification](https://mlsanity.com/paper/2206.09127/) (2022)
- [Topology Estimation of Simulated 4D Image Data by Combining Downscaling and Convolutional Neural Networks](https://mlsanity.com/paper/2306.14442/) (2023)
- [Topological Parallax: A Geometric Specification for Deep Perception Models](https://mlsanity.com/paper/2306.11835/) (2023)
- [Towards Scalable Topological Regularizers](https://mlsanity.com/paper/2501.14641/) (2025)
- [Convolutional Persistence Transforms](https://mlsanity.com/paper/2208.02107/) (2022)
- [A stability theorem for bigraded persistence barcodes](https://mlsanity.com/paper/2303.14694/) (2023)
- [Quotient Geometry and Persistence-Stable Metrics for Swarm Configurations](https://mlsanity.com/paper/2603.18041/) (2026)

## Abstract (arXiv)

What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient statistic. We therefore propose that the correct invariant is not the persistence diagram of X, but rather the collection of persistence diagrams of many small subsets. This invariant, which we call "distributed persistence," is perfectly parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of point clouds (with the quasi-isometry metric) to the space of distributed persistence invariants (with the Hausdorff-Bottleneck distance) is a global quasi-isometry. This is a much stronger property than simply being injective, as it implies that the inverse of a small neighborhood is a small neighborhood, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the quasi-isometry bounds depend on the size of the subsets taken, so that as the size of these subsets goes from small to large, the invariant interpolates between a purely geometric one and a topological one. Lastly, we note that our inverse results do not actually require considering all subsets of a fixed size (an enormous collection), but a relatively small collection satisfying certain covering properties that arise with high probability when randomly sampling subsets. These theoretical results are complemented by two synthetic experiments demonstrating the use of distributed persistence in practice.