Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares
Read the original on arXiv Machine Learning →The Flow has not summarised this story yet — read it at arXiv Machine Learning.
The Flow has not summarised this story yet — read it at arXiv Machine Learning.
The paper extends prior work on volume sampling by providing a Loewner envelope for the centered coefficient covariance in least‑squares regression with a fixed pool of features and responses. It characterizes when this envelope is tight, linking tightness to strict spectral properties of residuals, and introduces a residual‑augmented change of measure to derive a one‑sided slack bound. The results also offer geometric insights at the boundary and demonstrate non‑vacuous certificates through frozen‑feature examples, focusing on conditional centered, full‑Gram‑whitened covariance rather than population generalization.
The paper investigates volume‑sampled linear readouts with fixed feature pools and responses, focusing on the randomness introduced solely by subset selection. It establishes a globally sharp Loewner envelope for centered, full‑Gram‑whitened coefficient covariance and characterizes when a positive geometric margin exists versus when it vanishes, using conditions on residual covariance slack and a pairwise Naimark‑complement minor test. The results provide explicit geometric boundaries and conservative certificates for strictness and variance terms in fixed‑query squared loss, offering a design‑specific phase characterization for this randomized linear‑readout primitive.
We study ridge regression from exactly $s$ distinct rows of a fixed design. Responses are fixed, and only the subset is random. The determinant law and selected ridge fit share one positive definite p...
arXiv:2610.07721v1 Announce Type: cross Abstract: We study ridge regression from exactly $s$ distinct rows of a fixed design. Responses are fixed, and only the subset is random. The determinant law a...
The paper investigates restricted eigenvalue (RE) bounds for norm‑regularized estimators under heavy‑tailed designs. It shows that the previously conjectured sample‑size law based on Gaussian width fails for heavy‑tailed measurements, due to a phenomenon called simultaneous threshold occupancy. The authors provide explicit counterexamples, derive worst‑case sample‑complexity bounds, and compare the behavior of heavy‑tailed versus Gaussian designs on constant‑width polyhedral descent cones.
arXiv:2609.09130v1 Announce Type: new Abstract: An input may activate few hidden units even when different inputs collectively use an entire network. We study the statistical complexity of this input...