AI 中文总结
该研究针对含噪矩阵的零空间SVD估计，推导误差的紧致无穷阶表达式与级数形式，分析收敛半径与泛化风险，揭示谱混合相关的有限样本现象，为相关估计问题提供理论分析支撑。
AI 中文摘要
我们研究含噪矩阵的零空间估计问题。针对简单左零空间，我们首先推导最小左奇异向量误差的精确紧致表达式；随后给出SVD向量与投影子的全阶级数，再给出固定实现经验风险与条件总体泛化风险的紧致且一致截断级数形式。该递推可通过完整不变子空间扩展至多维零空间。收敛半径并非由误差图推断，而是独立于将保留特征值分支与其补集连接的最近复异常点计算得到。降零性实验表明，移动该谱边界可增大半径，但改进效果在保留零性上非单调。对于τ≥m的高斯训练下的各有序零方向，我们证明Wishart分裂矩阵W给出严格二阶经验排序；高斯平均在小噪声与极大噪声下均均衡主导泛化风险，而列交换定理证明各向同性信号子空间存在严格期望泛化排序。对于不等尖峰，精确总体重叠准则与99%蒙特卡洛置信证书解释观测到的中间排序。六阶风险修正改进了报告实验中的下交叉估计。这种等秩-等现象是与谱混合相关的有限样本诊断，但其容差交叉、异常点半径与渐近BBP阈值是三个不同的量。
英文摘要
We study null-space estimation from a noisy matrix. For a simple left null space, we first derive an exact compact expression for the error of the smallest left singular vector. We then give an all-order series for the SVD vector and projector, followed by compact and consistently truncated series forms for the fixed-realization empirical risk and conditional population generalization risk. The recursion extends to a multiple-dimensional null space by following the complete invariant subspace. The convergence radius is not inferred from an error plot: it is computed independently from the nearest complex exceptional point that joins a retained eigenvalue branch to its complement. A reduced-nullity experiment shows that moving this spectral boundary can increase the radius, although the improvement is not monotone in the retained nullity. For individually ordered null directions under Gaussian training with \(τ\geq m\), we prove that the Wishart splitting matrix \(W\) gives a strict second-order empirical ranking. Gaussian averaging equalizes the leading generalization risks at both small and very large noise, while a column-swap theorem proves strict expected generalization ranking for an isotropic signal subspace. For unequal spikes, an exact population-overlap criterion and a simultaneous \(99\%\) Monte Carlo confidence certificate explain the observed intermediate ranking. A sixth-order risk correction improves the lower-crossover estimate in the reported experiment. This equal--ranked--equal phenomenon is a finite-sample diagnostic related to spectral mixing, but its tolerance crossings, the exceptional-point radius, and the asymptotic BBP threshold are three distinct quantities.
Comments31 pages
