用于期望持久性图自适应向量化的Voronoi直方图
Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams
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中文总结 AI 辅助
本文提出用Voronoi直方图对期望持久性图（EPD）进行自适应向量化，建立了其稳定性界，在真实世界数据集的分类与降维任务中验证了该表示的有效性。
中文摘要 AI 辅助
众所周知，持久性图（PD）能有效捕捉点云拓扑结构，但其计算时间复杂度较高。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，是拓扑特征的一种分布。现有EPD向量化方法常依赖预定义的点变换，如高斯函数或景观函数。本文研究一种基于Voronoi直方图的替代离散化方法，用基于划分的自适应计数替代平滑函数近似，提出将基于Voronoi图的直方图用作EPD的向量化表示，不施加显式平滑点变换模型。在给定的分离与归一化条件下，本文建立了稳定性界，并明确该直方图表示何时能保留Wasserstein尺度的变化。在具有显著拓扑特征的真实世界数据集上，本文将所提表示应用于分类和降维任务，验证了其有效性。
英文摘要
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.
