Convexity-Driven Projection for Point Cloud Dimensionality Reduction

• URL: http://arxiv.org/abs/2509.22043v1
• Date: Fri, 26 Sep 2025 08:25:12 GMT
• Title: Convexity-Driven Projection for Point Cloud Dimensionality Reduction
• Authors: Suman Sanyal,
• Abstract summary: We propose Convexity-Driven detourile (CDP) for dimensionality reduction of point clouds.<n>CDP is a boundary-free linear method for dimensionality reduction of point clouds.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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