# Continuation of Point Clouds via Persistence Diagrams

arXiv: https://arxiv.org/abs/1506.03147 · HTML: https://mlsanity.com/paper/1506.03147/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2015.

## Built on by (most cited first, 13 of 13)

- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis](https://mlsanity.com/paper/2109.00530/) (2021)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [Optimisation of Spectral Wavelets for Persistence-based Graph Classification](https://mlsanity.com/paper/2101.05201/) (2021)
- [Topological Approach for Data Assimilation](https://mlsanity.com/paper/2411.18627/) (2024)
- [Convolutional Persistence Transforms](https://mlsanity.com/paper/2208.02107/) (2022)
- [Hall Effect Thruster Forecasting using a Topological Approach for Data Assimilation](https://mlsanity.com/paper/2504.06157/) (2025)
- [Dynamical System Parameter Path Optimization using Persistent Homology](https://mlsanity.com/paper/2505.00782/) (2025)
- [Non-negative Matrix Factorisation with Topological Regularisation](https://mlsanity.com/paper/2606.17531/) (2026)
- [A differentiability framework for zigzag persistent homology via linear interpolation](https://mlsanity.com/paper/2609.39242/) (2026)

## Abstract (arXiv)

In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton-Raphson continuation method in this setting. Given an original point cloud P, its persistence diagram D, and a target persistence diagram D', we gradually move from D to D', by successively computing intermediate point clouds until we finally find a point cloud P' having D' as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.