# Functional Summaries of Persistence Diagrams

arXiv: https://arxiv.org/abs/1804.01618 · HTML: https://mlsanity.com/paper/1804.01618/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2018.

## Built on by (most cited first, 19 of 19)

- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [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)
- [Topology-Aware Focal Loss for 3D Image Segmentation](https://mlsanity.com/paper/2304.12223/) (2023)
- [Topological Signatures vs. Gradient Histograms: A Comparative Study for Medical Image Classification](https://mlsanity.com/paper/2507.03006/) (2025)
- [HOG-CNN: Integrating Histogram of Oriented Gradients with Convolutional Neural Networks for Retinal Image Classification](https://mlsanity.com/paper/2507.22274/) (2025)
- [Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology](https://mlsanity.com/paper/2101.07752/) (2021)
- [Persistent Homology and Graphs Representation Learning](https://mlsanity.com/paper/2102.12926/) (2021)
- [Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set](https://mlsanity.com/paper/2106.00012/) (2021)
- [A computationally efficient framework for vector representation of persistence diagrams](https://mlsanity.com/paper/2109.08239/) (2021)
- [Data-driven and Automatic Surface Texture Analysis Using Persistent Homology](https://mlsanity.com/paper/2110.10005/) (2021)
- [A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering](https://mlsanity.com/paper/2105.12208/) (2021)
- [Transfer Learning for Autonomous Chatter Detection in Machining](https://mlsanity.com/paper/2204.05400/) (2022)
- [Vector Summaries of Persistence Diagrams for Permutation-based Hypothesis Testing](https://mlsanity.com/paper/2306.06257/) (2023)
- [Visualizing Topological Importance: A Class-Driven Approach](https://mlsanity.com/paper/2309.13185/) (2023)
- [Comparing the Effects of Persistence Barcodes Aggregation and Feature Concatenation on Medical Imaging](https://mlsanity.com/paper/2505.23637/) (2025)
- [Topology-Informed Machine Learning for Efficient Prediction of Solid Oxide Fuel Cell Electrode Polarization](https://mlsanity.com/paper/2410.05307/) (2024)
- [A Novel Patch-Based TDA Approach for Computed Tomography Imaging](https://mlsanity.com/paper/2512.12108/) (2025)
- [Test-Time Adaptation for Anomaly Segmentation via Topology-Aware Optimal Transport Chaining](https://mlsanity.com/paper/2601.20333/) (2026)
- [Learning Topology-Aware Representations via Test-Time Adaptation for Anomaly Segmentation](https://mlsanity.com/paper/2606.28268/) (2026)

## Abstract (arXiv)

One of the primary areas of interest in applied algebraic topology is persistent homology, and, more specifically, the persistence diagram. Persistence diagrams have also become objects of interest in topological data analysis. However, persistence diagrams do not naturally lend themselves to statistical goals, such as inferring certain population characteristics, because their complicated structure makes common algebraic operations--such as addition, division, and multiplication-- challenging (e.g., the mean might not be unique). To bypass these issues, several functional summaries of persistence diagrams have been proposed in the literature (e.g. landscape and silhouette functions). The problem of analyzing a set of persistence diagrams then becomes the problem of analyzing a set of functions, which is a topic that has been studied for decades in statistics. First, we review the various functional summaries in the literature and propose a unified framework for the functional summaries. Then, we generalize the definition of persistence landscape functions, establish several theoretical properties of the persistence functional summaries, and demonstrate and discuss their performance in the context of classification using simulated prostate cancer histology data, and two-sample hypothesis tests comparing human and monkey fibrin images, after developing a simulation study using a new data generator we call the Pickup Sticks Simulator (STIX).