Compact Screens and Pyramidal Compactification of Geometric Data Sets
Abstract
We introduce an observable distance that compares realvalued features through a fixed bounded coordinate. The coordinate retains the distinction between finite feature values while compressing their independent escape to infinity, the source of nonseparability for the classical observable distance on all geometric data sets. The new distance makes the full class separable and geodesic: every pair is joined by a constant-speed path, and on metric measure spaces the induced topology agrees with the concentration topology. From the same coordinate we construct compact screens, whose features take values in one fixed compact interval. The screened class is Polish and geodesic for the Box distance. Organizing its finite-feature quotients by the feature order yields a compact pyramid space. Pyramids generated by single compact screens form a dense subspace, so this pyramid space compactifies the original class after passage to compact screens. Convergence is detected through the Box-Hausdorff behavior of every finite measurement layer. Finally, taking sum-metric products with a common metric measure factor is nonexpansive for the new distance. More precisely, the comparison determined by a prescribed coupling of the original spaces and the diagonal coupling of the common factor is preserved, whereas optimization over all product couplings yields the nonexpansive inequality.
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BibTeX
@misc{yokota2026compact,
title = {{Compact Screens and Pyramidal Compactification of Geometric Data Sets}},
author = {Shigeaki Yokota},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26262},
eprint = {2609.26262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26262},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compact Screens and Pyramidal Compactification of Geometric Data Sets
AU - Shigeaki Yokota
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.26262
UR - https://arxiv.org/abs/2609.26262
AB - We introduce an observable distance that compares realvalued features through a fixed bounded coordinate. The coordinate retains the distinction between finite feature values while compressing their independent escape to infinity, the source of nonseparability for the classical observable distance on all geometric data sets. The new distance makes the full class separable and geodesic: every pair is joined by a constant-speed path, and on metric measure spaces the induced topology agrees with the concentration topology. From the same coordinate we construct compact screens, whose features take values in one fixed compact interval. The screened class is Polish and geodesic for the Box distance. Organizing its finite-feature quotients by the feature order yields a compact pyramid space. Pyramids generated by single compact screens form a dense subspace, so this pyramid space compactifies the original class after passage to compact screens. Convergence is detected through the Box-Hausdorff behavior of every finite measurement layer. Finally, taking sum-metric products with a common metric measure factor is nonexpansive for the new distance. More precisely, the comparison determined by a prescribed coupling of the original spaces and the diagonal coupling of the common factor is preserved, whereas optimization over all product couplings yields the nonexpansive inequality.
ER -