参数化最小二乘问题的不变性指标及其意义
The Index of Invariance and its Implications for a Parameterized Least Squares Problem
摘要：
我们研究了问题$x_{b,ω} := \text{arg min}_{x \in \mathcal{S}} \|(A + ωI)^{-1/2} (b - Ax)\|_2$，与$A = A^*$，一个子空间$\mathcal{S}$的$\mathbb{F}^n$（$\mathbb{F} = \mathbb{R}$或$\mathbb{C}$），和$ω> -λ_{min}(A)$。我们发现，存在一个子空间$\mathcal{Y}$的$\mathbb{F}^n$，独立的$b$，使$\{x_{b,ω} - x_{b,μ} \mid ω,μ> -λ_{min}(A)\} \subseteq \mathcal{Y}$，其中$\dim(\mathcal{Y}) \leq \dim(\mathcal{S} + A\mathcal{S}) - \dim(\mathcal{S}) = \mathbf{Ind}_A(\mathcal{S})$，一个数量，我们称之为指数的不变性的$\mathcal{S}$关于$A$。特别是如果$\mathcal{S}$是Krylov子空间，这意味着Hallman & Gu（2018）的低维结果。问题也是这样的，当$A$是积极的和$\mathcal{S}$是Krylov子空间，它减少到CG的$ω= 0$和MINRES的$ω\to \infty$。我们研究了$\mathbf{Ind}_A(\mathcal{S})$与$A$和$\mathcal{S}$的关系。我们证明了包含解$x_{b,ω}$的仿射子空间$\mathcal{X}_b$的维数可以小于$\mathbf{Ind}_A(\mathcal{S})$，对于所有的$b$。然而，我们也给出了$A$和$\mathcal{S}$的一些充分条件，在这些条件下，$\mathcal{X} := \text{Span}{\{x_{b,ω} - x_{b,μ} \mid b \in \mathbb{F}^n, ω,μ> -λ_{min}(A)\}}$的维数等于$\mathbf{Ind}_A(\mathcal{S})$。然后，我们研究了映射$ω\mapsto x_{b,ω}$的内射性，从而证明了Hallman & Gu（2018）的凸性结果。最后，我们展示了嵌套子空间$\mathcal{S} \subseteq \mathcal{S}' \subseteq \mathbb{F}^n$的集合，如$M(\mathcal{S},\mathcal{S}') = \{A \in \mathbb{F}^{n \times n} \mid \mathcal{S} + A\mathcal{S} = \mathcal{S}'\}$，形成光滑的实流形，并探讨了它们之间的一些拓扑关系。
英文摘要：
We study the problem $x_{b,ω} := \text{arg min}_{x \in \mathcal{S}} \|(A + ωI)^{-1/2} (b - Ax)\|_2$, with $A = A^*$, for a subspace $\mathcal{S}$ of $\mathbb{F}^n$ ($\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$), and $ω> -λ_{min}(A)$. We show that there exists a subspace $\mathcal{Y}$ of $\mathbb{F}^n$, independent of $b$, such that $\{x_{b,ω} - x_{b,μ} \mid ω,μ> -λ_{min}(A)\} \subseteq \mathcal{Y}$, where $\dim(\mathcal{Y}) \leq \dim(\mathcal{S} + A\mathcal{S}) - \dim(\mathcal{S}) = \mathbf{Ind}_A(\mathcal{S})$, a quantity which we call the index of invariance of $\mathcal{S}$ with respect to $A$. In particular if $\mathcal{S}$ is a Krylov subspace, this implies the low dimensionality result of Hallman & Gu (2018). The problem is also such that when $A$ is positive and $\mathcal{S}$ is a Krylov subspace, it reduces to CG for $ω= 0$ and to MINRES for $ω\to \infty$. We study several properties of $\mathbf{Ind}_A(\mathcal{S})$ in relation to $A$ and $\mathcal{S}$. We show that the dimension of the affine subspace $\mathcal{X}_b$ containing the solutions $x_{b,ω}$ can be smaller than $\mathbf{Ind}_A(\mathcal{S})$ for all $b$. However, we also exhibit some sufficient conditions on $A$ and $\mathcal{S}$, under which $\mathcal{X} := \text{Span}{\{x_{b,ω} - x_{b,μ} \mid b \in \mathbb{F}^n, ω,μ> -λ_{min}(A)\}}$ has dimension equal to $\mathbf{Ind}_A(\mathcal{S})$. We then study the injectivity of the map $ω\mapsto x_{b,ω}$, leading us to a proof of the convexity result from Hallman & Gu (2018). We finish by showing that sets such as $M(\mathcal{S},\mathcal{S}') = \{A \in \mathbb{F}^{n \times n} \mid \mathcal{S} + A\mathcal{S} = \mathcal{S}'\}$, for nested subspaces $\mathcal{S} \subseteq \mathcal{S}' \subseteq \mathbb{F}^n$, form smooth real manifolds, and explore some topological relationships between them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
