# Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams

arXiv: https://arxiv.org/abs/2607.27126 · HTML: https://mlsanity.com/paper/2607.27126/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 68 references that are in the map.
- First posted on arXiv 29 Jul 2026.

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [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)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Stochastic Convergence of Persistence Landscapes and Silhouettes](https://mlsanity.com/paper/1312.0308/) (2013)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [Persistence Curves: A canonical framework for summarizing persistence diagrams](https://mlsanity.com/paper/1904.07768/) (2019)
- [Kernel method for persistence diagrams via kernel embedding and weight factor](https://mlsanity.com/paper/1706.03472/) (2017)
- [The persistence landscape and some of its properties](https://mlsanity.com/paper/1810.04963/) (2018)
- [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)
- [A persistence landscapes toolbox for topological statistics](https://mlsanity.com/paper/1501.00179/) (2014)
- [The density of expected persistence diagrams and its kernel based estimation](https://mlsanity.com/paper/1802.10457/) (2018)
- [Approximating Continuous Functions on Persistence Diagrams Using Template Functions](https://mlsanity.com/paper/1902.07190/) (2019)
- [Comparing persistence diagrams through complex vectors](https://mlsanity.com/paper/1505.01335/) (2015)
- [Estimation and Quantization of Expected Persistence Diagrams](https://mlsanity.com/paper/2105.04852/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds](https://mlsanity.com/paper/2307.09259/) (2023)
- [GEFL: Extended Filtration Learning for Graph Classification](https://mlsanity.com/paper/2406.02732/) (2024)
- [Adaptive template systems: Data-driven feature selection for learning with persistence diagrams](https://mlsanity.com/paper/1910.06741/) (2019)
- [Persistent Homology for Breast Tumor Classification using Mammogram Scans](https://mlsanity.com/paper/2201.02295/) (2022)
- [Persistence Bag-of-Words for Topological Data Analysis](https://mlsanity.com/paper/1812.09245/) (2018)

## Abstract (arXiv)

Persistence Diagram (PD) is known to capture point cloud topology effectively, but its computation has high time complexity. Expected Persistence Diagram (EPD) has been developed to reduce the time cost by studying the topology of multiple subsets of a point cloud and it serves as a distribution of topological features. Existing EPD vectorizations often rely on predefined point transformations, such as Gaussian or landscape functions. We study an alternative discretization based on Voronoi histograms, which trades smooth functional approximation for adaptive partition-based counting. We propose to use Voronoi Diagram-based histogram as the vectorization of EPD, without imposing an explicit smooth point transformation model. Under stated separation and normalization conditions, we establish stability bounds and characterize when the histogram representation preserves Wasserstein-scale variation. We demonstrate the effectiveness of our proposed representation on real-world datasets which have significant topological features for classification and dimensionality reduction tasks.