何时Sharp协方差包络是紧的？体积采样最小二乘的仅特征几何
Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares
AI总结：
该研究针对普通体积采样结合选定未加权最小二乘的场景，建立了中心化系数协方差的Loewner包络，分析了其Sharp性、可达性及相关几何性质，明确了证书非空的条件。
AI中文摘要：
Derezinski和Warmuth的先前分析建立了普通体积采样的全尺寸采样恒等式、选定普通最小二乘（OLS）的无偏性以及逆矩，而他们的任意固定响应损失和预测协方差的精确公式处于秩-大小端点s=d。我们针对普通索引固定大小体积采样后接选定未加权最小二乘的情况，在每个满秩固定池、响应和合法预算d≤s≤m下，建立了中心化系数协方差的Loewner包络；其系数在满秩类上是全局Sharp的。全局Sharp性不决定当前池上的可达性。在正损失、严格内部预算且无 coloops 的条件下，仅特征余量ν_A给出精确的固定设计谱相位：当且仅当每个相容残差的归一化谱包络是严格的时，ν_A>0；当且仅当某个相容残差是谱紧的时，ν_A=0；该零余量残差在每个严格内部预算下都是紧的。残差增强的测度变化提供了响应感知机制和单侧定量松弛界，而支撑饱和证明了可达方向。关键的等杠杆几何解释了边界，可靠的下界证书产生了保守的同原语基数决策。冻结特征示例表明该证书非空，并测量其授权约简的固定池成本。这些结论涉及条件中心化、全Gram白化系数协方差，而非总体泛化。
英文摘要：
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.
