分析噪声下矩阵分解的稳定性，给出条件数的概率上界。
Probabilistic Analysis of Least Squares, Orthogonal Projection, and QR Factorization Algorithms Subject to Gaussian Noise
- 基于高斯噪声建模，推导投影残差与范数的精确概率分布。
- 在不完美正交化条件下，给出QR算法条件数的概率上界。
- 适合研究数值稳定性和误差传播的算法工程师参考。
本文研究高斯扰动对最小二乘残差、正交投影及QR型算法的影响。核心问题为：给定一个满列秩矩阵 $B\in\mathbb{R}^{m\times n}$，若向其添加一个归一化列 $q=(x+y)/\|x+y\|_2$，其中 $x\perp\operatorname{span}(B)$ 为理想正交分量，$y$ 为正交化误差，那么新矩阵 $[B,q]$ 的条件数 $κ([B,q])$ 最大可能达到多少？我们给出了基于 $B$ 的极值奇异值和 $\|B^T y\|_2/\|x+y\|_2$ 的Weyl型奇异值界，并进一步推导了在高斯扰动下范数与投影残差的精确概率分布。利用这些分布，我们得到了在不完美正交化与精确归一化条件下QR型过程的条件数概率上界。
原文摘要 · Abstract (English)
We consider the effect of Gaussian perturbations on least-squares residuals, orthogonal projections, and QR-type algorithms. The problem that motivated our investigations is as follows: suppose that a full column-rank matrix \(B\in\mathbb{R}^{m\times n}\) has already been computed, and suppose that a new normalized column \(q=(x+y)/\|x+y\|_2\) is to be appended to \(B\), where \(x\perp\operatorname{span}(B)\) is the ideal orthogonal component and \(y\) represents the orthogonalization error. How large can the condition number \(κ([B,q])\) of the resulting matrix \([B,q]\) become? While we provide a Weyl-type bound on the singular values of \([B,q]\), in terms of the extremal singular values of \(B\) and the quantity \(\|B^T y\|_2/\|x+y\|_2\), we also derive exact probability laws for norms and projection residuals under Gaussian perturbations. Finally, we use these probability laws to derive probabilistic condition-number bounds for QR-type processes with imperfect orthogonalization and exact normalization.
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