提出一种无边界点云降维方法，保留局部非凸结构。
Convexity-Driven Projection for Point Cloud Dimensionality Reduction
- 基于邻近图识别低欧氏-最短路径比的点对，构建非凸性矩阵。
- 投影使用矩阵前k个特征向量，保证点对间畸变上界可验证。
- 提供可检查的误差量化指标，适合关注降维可靠性的人群。
我们提出凸性驱动投影（CDP），一种无边界的线性点云降维方法，旨在保留由绕行引发的局部非凸性。CDP 构建 $k$-NN 图，识别欧氏距离与最短路径比率低于阈值的可接受点对，并将其归一化方向聚合为半正定的非凸性结构矩阵。投影采用该矩阵的前 $k$ 个特征向量。本文给出两个可验证的保证：一对后验证书，用于限定每个可接受点对在投影后的畸变；以及平均情况下的谱界，将捕获的方向能量与结构矩阵的谱关联，得出典型畸变的分位数结论。评估协议报告固定对与重选对的绕行误差及证书分位数，使从业者可在其数据上验证保证。
原文摘要 · Abstract (English)
We propose Convexity-Driven Projection (CDP), a boundary-free linear method for dimensionality reduction of point clouds that targets preserving detour-induced local non-convexity. CDP builds a $k$-NN graph, identifies admissible pairs whose Euclidean-to-shortest-path ratios are below a threshold, and aggregates their normalized directions to form a positive semidefinite non-convexity structure matrix. The projection uses the top-$k$ eigenvectors of the structure matrix. We give two verifiable guarantees. A pairwise a-posteriori certificate that bounds the post-projection distortion for each admissible pair, and an average-case spectral bound that links expected captured direction energy to the spectrum of the structure matrix, yielding quantile statements for typical distortion. Our evaluation protocol reports fixed- and reselected-pairs detour errors and certificate quantiles, enabling practitioners to check guarantees on their data.
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