Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams

2026-07-29

AI Summary

Existing vectorization methods for expected persistence diagrams (EPDs) rely on predefined smoothing transformations, which struggle to adaptively capture the distribution of topological features. This work proposes a Voronoi diagram–based histogram vectorization approach that achieves adaptive discretization of EPDs through spatial partitioning and counting, eliminating the need for explicit smoothing functions. To the best of our knowledge, this is the first method to incorporate Voronoi histograms into EPD representations, offering theoretical guarantees of stability and information preservation under the Wasserstein metric under certain conditions. Experimental results demonstrate that the proposed representation effectively captures essential topological features and outperforms conventional vectorization techniques in classification and dimensionality reduction tasks on real-world datasets.

Abstract

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.

Problem

• Expected Persistence Diagram
• Vectorization
• Voronoi Histogram
• Topological Data Analysis
• Wasserstein Stability

Innovation

• Voronoi Histogram
• Expected Persistence Diagram
• Adaptive Vectorization
• Topological Data Analysis
• Wasserstein Stability