# Persistence Images: A Stable Vector Representation of Persistent Homology

arXiv: https://arxiv.org/abs/1507.06217 · HTML: https://mlsanity.com/paper/1507.06217/

## Key facts

- Cited by 236 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2015.

## Built on by (most cited first, 30 of 236)

- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Z-GCNETs: Time Zigzags at Graph Convolutional Networks for Time Series Forecasting](https://mlsanity.com/paper/2105.04100/) (2021)
- [Trigger Hunting with a Topological Prior for Trojan Detection](https://mlsanity.com/paper/2110.08335/) (2021)
- [Link Prediction with Persistent Homology: An Interactive View](https://mlsanity.com/paper/2102.10255/) (2021)
- [Topological Detection of Trojaned Neural Networks](https://mlsanity.com/paper/2106.06469/) (2021)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [TransFace++: Rethinking the Face Recognition Paradigm with a Focus on Accuracy, Efficiency, and Security](https://mlsanity.com/paper/2308.10133/) (2023)
- [RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds](https://mlsanity.com/paper/2202.01725/) (2022)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [Curvature Filtrations for Graph Generative Model Evaluation](https://mlsanity.com/paper/2301.12906/) (2023)
- [On the Expressivity of Persistent Homology in Graph Learning](https://mlsanity.com/paper/2302.09826/) (2023)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Neural Approximation of Graph Topological Features](https://mlsanity.com/paper/2201.12032/) (2022)
- [Topological Neural Networks go Persistent, Equivariant, and Continuous](https://mlsanity.com/paper/2406.03164/) (2024)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [Topological Attention for Time Series Forecasting](https://mlsanity.com/paper/2107.09031/) (2021)
- [TopoGCL: Topological Graph Contrastive Learning](https://mlsanity.com/paper/2406.17251/) (2024)
- [Entropy Partial Transport with Tree Metrics: Theory and Practice](https://mlsanity.com/paper/2101.09756/) (2021)
- [TopoTxR: A Topological Biomarker for Predicting Treatment Response in Breast Cancer](https://mlsanity.com/paper/2105.06049/) (2021)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning](https://mlsanity.com/paper/2304.04970/) (2023)
- [Topology-Aware Latent Diffusion for 3D Shape Generation](https://mlsanity.com/paper/2401.17603/) (2024)
- [TDA-Net: Fusion of Persistent Homology and Deep Learning Features for COVID-19 Detection in Chest X-Ray Images](https://mlsanity.com/paper/2101.08398/) (2021)
- [Tensor-view Topological Graph Neural Network](https://mlsanity.com/paper/2401.12007/) (2024)
- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)
- [From Geometry to Topology: Inverse Theorems for Distributed Persistence](https://mlsanity.com/paper/2101.12288/) (2021)
- [The Topological BERT: Transforming Attention into Topology for Natural Language Processing](https://mlsanity.com/paper/2206.15195/) (2022)

## Abstract (arXiv)

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.