# Discrete transforms of quantized persistence diagrams

arXiv: https://arxiv.org/abs/2312.17093 · HTML: https://mlsanity.com/paper/2312.17093/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 49 references that are in the map.
- First posted on arXiv 28 Dec 2023.

## Builds on (most cited first, 18 of 18)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [RetGK: Graph Kernels based on Return Probabilities of Random Walks](https://mlsanity.com/paper/1809.02670/) (2018)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [Uncovering the Topology of Time-Varying fMRI Data using Cubical Persistence](https://mlsanity.com/paper/2006.07882/) (2020)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport](https://mlsanity.com/paper/1901.03048/) (2019)
- [Supervised learning of sheared distributions using linearized optimal transport](https://mlsanity.com/paper/2201.10590/) (2022)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [The Morse theory of vCech and Delaunay complexes](https://mlsanity.com/paper/1312.1231/) (2013)
- [Statistical learning on measures: an application to persistence diagrams](https://mlsanity.com/paper/2303.08456/) (2023)
- [Scale-variant topological information for characterizing the structure of complex networks](https://mlsanity.com/paper/1811.03573/) (2018)
- [Musical Stylistic Analysis: A Study of Intervallic Transition Graphs via Persistent Homology](https://mlsanity.com/paper/2204.11139/) (2022)

## Built on by (most cited first, 1 of 1)

- [Hierarchical Clustering in LambdaCDM Cosmologies via Persistence Energy](https://mlsanity.com/paper/2401.01988/) (2024)

## Abstract (arXiv)

Topological data analysis leverages topological features to analyze datasets, with applications in diverse fields like medical sciences and biology. A key tool of this theory is the persistence diagram, which encodes topological information but poses challenges for integration into standard machine learning pipelines. We introduce Qupid (QUantized Persistence and Integral transforms of Diagrams), a novel and simple method for vectorizing persistence diagrams. First, Qupid uses a binning procedure to turn persistence diagrams into finite measures on a grid and then applies discrete transforms to these measures. Key features are the choice of log-scaled grids that emphasize information contained near the diagonal in persistence diagrams, combined with the use of discrete transforms to enhance and efficiently encode the obtained topological information. We conduct an in-depth experimental analysis of Qupid, showing that the simplicity of our method results in very low computational costs while preserving highly competitive performances compared to state-of-the-art methods across numerous classification tasks on both synthetic and real-world datasets. Finally, we provide experimental evidence that our method is robust to a decrease in the grid resolution used.