Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams
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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.
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References · 32
- 2017Attention Is All You Needcited by 60,361
- 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,248
- 2017Deep Setscited by 1,421
- 2018The UCR Time Series Archivecited by 431
- 2015Persistence Images: A Stable Vector Representation of Persistent Homologycited by 236
- 2017An introduction to Topological Data Analysis: fundamental and practical aspects for data scientistscited by 172
- 2019PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signaturescited by 126
- 2017Deep Learning with Topological Signaturescited by 114
- 2021Topological Graph Neural Networkscited by 87
- 2017Sliced Wasserstein Kernel for Persistence Diagramscited by 77
- 2014A Stable Multi-Scale Kernel for Topological Machine Learningcited by 75
- 2019Graph Filtration Learningcited by 69
20 more
- 2013Stochastic Convergence of Persistence Landscapes and Silhouettescited by 43
- 2018Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagramscited by 33
- 2019Persistence Curves: A canonical framework for summarizing persistence diagramscited by 33
- 2017Kernel method for persistence diagrams via kernel embedding and weight factorcited by 20
- 2018The persistence landscape and some of its propertiescited by 19
- 2014Subsampling Methods for Persistent Homologycited by 17
- 2018Persistence paths and signature features in topological data analysiscited by 17
- 2014A persistence landscapes toolbox for topological statisticscited by 16
- 2018The density of expected persistence diagrams and its kernel based estimationcited by 13
- 2019Approximating Continuous Functions on Persistence Diagrams Using Template Functionscited by 10
- 2015Comparing persistence diagrams through complex vectorscited by 9
- 2021Estimation and Quantization of Expected Persistence Diagramscited by 8
- 2022Approximating Persistent Homology for Large Datasetscited by 8
- 2023Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Cloudscited by 8
- 2024GEFL: Extended Filtration Learning for Graph Classificationcited by 5
- 2019Adaptive template systems: Data-driven feature selection for learning with persistence diagramscited by 4
- 2022Persistent Homology for Breast Tumor Classification using Mammogram Scanscited by 4
- 2018Persistence Bag-of-Words for Topological Data Analysiscited by 3
- 2019Persistence B-Spline Grids: Stable Vector Representation of Persistence Diagrams Based on Data Fittingcited by 3
- 2024Multiset Transformer: Advancing Representation Learning in Persistence Diagramscited by 2
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