arXiv：cs.LG· Kihun Rhee, Hanjoon Byun, Junpyo Seo·· 4 小时前AI 评分30
体积采样最小二乘中的接触几何与协方差亏损
Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares
AI 导读
该研究刻画了固定规模体积采样后无权重最小二乘在固定设计上达到系数协方差上界的条件：对无列缺失的实白化设计，接触空间在每个严格内点样本量下保持不变，非零取值构成有限正交族。证明基于归一化协方差亏损与留一算子的双边 Loewner 比较，并给出期望固定查询平方损失超额的上下界。论文 55 页，含查询风险界、精确构造与算术校验。
正文
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.
| Comments: | 55 pages. Revised title and substantially reorganized theoretical exposition. Includes contact-space and whole-query classification, all-size leave-one-out covariance-deficit comparisons, quantitative contact geometry, and query-risk bounds, with proofs, exact constructions, and ancillary exact-arithmetic checks |
| Subjects: | Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2608.26877 [cs.LG] |
| (or arXiv:2608.26877v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2608.26877 arXiv-issued DOI via DataCite |
Submission history
From: Kihun Rhee [view email]
[v1]
Thu, 27 Aug 2026 09:38:54 UTC (476 KB)
[v2]
Tue, 8 Sep 2026 03:47:43 UTC (477 KB)
[v3]
Tue, 6 Oct 2026 01:32:35 UTC (85 KB)
来源：arXiv：cs.LG · arxiv.org