# Cycle Registration in Persistent Homology with Applications in Topological Bootstrap

arXiv: https://arxiv.org/abs/2101.00698 · HTML: https://mlsanity.com/paper/2101.00698/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 80 references that are in the map.
- First posted on arXiv 3 Jan 2021.

## Builds on (most cited first, 13 of 13)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Fast and Accurate Tumor Segmentation of Histology Images using Persistent Homology and Deep Convolutional Features](https://mlsanity.com/paper/1805.03699/) (2018)
- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Beyond Deep Residual Learning for Image Restoration: Persistent Homology-Guided Manifold Simplification](https://mlsanity.com/paper/1611.06345/) (2016)
- [Confidence sets for persistence diagrams](https://mlsanity.com/paper/1303.7117/) (2013)
- [Subsampling Methods for Persistent Homology](https://mlsanity.com/paper/1406.1901/) (2014)
- [Persistence paths and signature features in topological data analysis](https://mlsanity.com/paper/1806.00381/) (2018)
- [Modeling and replicating statistical topology, and evidence for CMB non-homogeneity](https://mlsanity.com/paper/1704.08248/) (2017)

## Built on by (most cited first, 8 of 8)

- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Fast Topological Signal Identification and Persistent Cohomological Cycle Matching](https://mlsanity.com/paper/2209.15446/) (2022)
- [Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds](https://mlsanity.com/paper/2402.15058/) (2024)
- [Topological Quality of Subsets via Persistence Matching Diagrams](https://mlsanity.com/paper/2306.02411/) (2023)
- [Hybridization of Persistent Homology with Neural Networks for Time-Series Prediction: A Case Study in Wave Height](https://mlsanity.com/paper/2409.01519/) (2024)
- [Cycles Communities from the Perspective of Dendrograms and Gradient Sampling](https://mlsanity.com/paper/2512.12974/) (2025)

## Abstract (arXiv)

In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes, along with suitable metrics (e.g. Wasserstein). These summaries are useful for computational purposes, but they are merely a marginal of the actual topological information that persistent homology can provide. Instead, our approach compares between two topological representations directly in the data space. We do so by defining a correspondence relation between individual persistent cycles of two different spaces, and devising a method for computing this correspondence. Our matching of cycles is based on both the persistence intervals and the spatial placement of each feature. We demonstrate our new framework in the context of topological inference, where we use statistical bootstrap methods in order to differentiate between real features and noise in point cloud data.