Compact and Infinite-Order Error Analysis for Null-Space SVD Estimation

2026-08-31

AI Summary

本文研究了噪声矩阵的零空间估计问题，通过推导精确的紧致表达式和全阶级数展开方法来分析SVD向量及投影误差，并探讨了不同条件下的风险评估。

Abstract

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.

Problem

• null-space estimation
• singular vector error
• SVD
• convergence radius
• generalization risk

Innovation

• Compact Error Expression
• All-Order Series for SVD Vector and Projector
• Convergence Radius Computation via Exceptional Points
• Wishart Splitting Matrix
• Population-Overlap Criterion