# Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares

arXiv: https://arxiv.org/abs/2608.26877 · HTML: https://mlsanity.com/paper/2608.26877/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 13 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 1 of 1)

- [Adaptive randomized pivoting and volume sampling](https://mlsanity.com/paper/2510.02513/) (2025)

## Built on by (most cited first, 1 of 1)

- [Exact Calibration and Sharp Risk Geometry for Volume-Sampled Ridge Regression](https://mlsanity.com/paper/2610.07721/) (2026)

## Abstract (arXiv)

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.