# Topology Applied to Machine Learning: From Global to Local

arXiv: https://arxiv.org/abs/2103.05796 · HTML: https://mlsanity.com/paper/2103.05796/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 81 references that are in the map.
- First posted on arXiv 10 Mar 2021.

## Builds on (most cited first, 22 of 22)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Learning metrics for persistence-based summaries and applications for graph classification](https://mlsanity.com/paper/1904.12189/) (2019)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [Topological Data Analysis of Decision Boundaries with Application to Model Selection](https://mlsanity.com/paper/1805.09949/) (2018)
- [Confidence sets for persistence diagrams](https://mlsanity.com/paper/1303.7117/) (2013)
- [Using persistent homology and dynamical distances to analyze protein binding](https://mlsanity.com/paper/1412.1394/) (2014)
- [Persistent Homology and the Upper Box Dimension](https://mlsanity.com/paper/1802.00533/) (2018)
- [Embeddings of Persistence Diagrams into Hilbert Spaces](https://mlsanity.com/paper/1905.05604/) (2019)
- [Hypothesis Testing for Topological Data Analysis](https://mlsanity.com/paper/1310.7467/) (2013)
- [Comparing persistence diagrams through complex vectors](https://mlsanity.com/paper/1505.01335/) (2015)
- [On the choice of weight functions for linear representations of persistence diagrams](https://mlsanity.com/paper/1807.03678/) (2018)
- [Statistical Analysis of Persistence Intensity Functions](https://mlsanity.com/paper/1510.02502/) (2015)
- [Capturing Dynamics of Time-Varying Data via Topology](https://mlsanity.com/paper/2010.05780/) (2020)
- [Topological descriptors for 3D surface analysis](https://mlsanity.com/paper/1601.06057/) (2016)
- [Topological and Statistical Behavior Classifiers for Tracking Applications](https://mlsanity.com/paper/1406.0214/) (2014)
- [Machine learning with persistent homology and chemical word embeddings improves prediction accuracy and interpretability in metal-organic frameworks](https://mlsanity.com/paper/2010.00532/) (2020)

## Built on by (most cited first, 4 of 4)

- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [TopoGCL: Topological Graph Contrastive Learning](https://mlsanity.com/paper/2406.17251/) (2024)
- [Singular value decomposition based matrix surgery](https://mlsanity.com/paper/2302.11446/) (2023)
- [Can Persistent Homology provide an efficient alternative for Evaluation of Knowledge Graph Completion Methods?](https://mlsanity.com/paper/2301.12929/) (2023)

## Abstract (arXiv)

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning.