{"ID":2865153,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.22043","arxiv_id":"2509.22043","title":"Convexity-Driven Projection for Point Cloud Dimensionality Reduction","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.","short_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...","url_abs":"https://arxiv.org/abs/2509.22043","url_pdf":"https://arxiv.org/pdf/2509.22043v1","authors":"[\"Suman Sanyal\"]","published":"2025-09-26T08:25:12Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"stat.ML\"]","methods":"[]","has_code":false}