Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares
Abstract
We classify when ordinary fixed-size volume sampling followed by unweighted least squares attains its sharp coefficient-covariance ceiling on a fixed design. For a real whitened design without coloops and a fixed positive-loss residual, the contact space is unchanged at every strict-interior sample size. Its possible nonzero values form a finite orthogonal family: each maximal parallel class of normalized Naimark-complement rows determines a deletion nullspace of dimension one less than the class size. A single residual attains an entire query precisely when the query range lies in one class space. The proof starts from two-sided Loewner comparison of every normalized covariance deficit with an explicit leave-one-out operator, using supported omission moments and reverse deletion. Residual augmentation provides resolvent and second-moment upper bounds, while complement geometry yields query-specific margins, angular concentration, local alignment, and a multi-output energy obstruction. Exact families give closed-form margins and covariances, exhibit support-boundary jumps, and approach the ceiling despite a uniformly positive geometric margin. Finally, the same moment identities give upper and lower bounds on expected fixed-query squared-loss excess. The subset draw is the only randomness; all support and endpoint restrictions are explicit.
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BibTeX
@misc{rhee2026contact,
title = {{Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares}},
author = {Kihun Rhee and Hanjoon Byun and Junpyo Seo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.26877},
eprint = {2608.26877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26877},
}
RIS
TY - UNPB
TI - Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares
AU - Kihun Rhee
AU - Hanjoon Byun
AU - Junpyo Seo
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2608.26877
UR - https://arxiv.org/abs/2608.26877
AB - We classify when ordinary fixed-size volume sampling followed by unweighted least squares attains its sharp coefficient-covariance ceiling on a fixed design. For a real whitened design without coloops and a fixed positive-loss residual, the contact space is unchanged at every strict-interior sample size. Its possible nonzero values form a finite orthogonal family: each maximal parallel class of normalized Naimark-complement rows determines a deletion nullspace of dimension one less than the class size. A single residual attains an entire query precisely when the query range lies in one class space. The proof starts from two-sided Loewner comparison of every normalized covariance deficit with an explicit leave-one-out operator, using supported omission moments and reverse deletion. Residual augmentation provides resolvent and second-moment upper bounds, while complement geometry yields query-specific margins, angular concentration, local alignment, and a multi-output energy obstruction. Exact families give closed-form margins and covariances, exhibit support-boundary jumps, and approach the ceiling despite a uniformly positive geometric margin. Finally, the same moment identities give upper and lower bounds on expected fixed-query squared-loss excess. The subset draw is the only randomness; all support and endpoint restrictions are explicit.
ER -