用沃罗诺伊直方图自适应向量化期望拓扑图，提升计算效率与稳定性。
Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams
- 基于沃罗诺伊图的自适应分区计数替代传统光滑函数转换
- 在特定条件下实现与沃尔什距离尺度一致的稳定表示
- 适用于需要拓扑特征分类与降维的真实数据集
持久性图（PD）能有效捕捉点云拓扑结构，但其计算复杂度高。期望持久性图（EPD）通过研究点云多个子集的拓扑结构来降低时间成本，作为拓扑特征的分布表示。现有EPD向量化方法通常依赖预定义的点变换，如高斯或景观函数。本文提出基于沃罗诺伊直方图的替代离散化方法，以自适应分块计数取代光滑函数逼近。在给定分离与归一化条件下，建立稳定性界，并刻画了直方图表示如何保持沃尔什距离尺度的变异性。在具有显著拓扑特征的真实数据集上，验证了该表示在分类与降维任务中的有效性。
原文摘要 · Abstract (English)
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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