论文标题
不变性索引及其对参数化最小二乘问题的影响
The Index of Invariance and its Implications for a Parameterized Least Squares Problem
论文作者
论文摘要
我们研究问题$ x_ {b，ω}：= \ text {arg min} _ {x \ in \ Mathcal {s}}}} \ |（a +ωi）^{ - 1/2}（b- ax）（b -ax）\ | _2 $ $ \ mathbb {f}^n $（$ \ mathbb {f} = \ mathbb {r} $或$ \ mathbb {c} $）和$ω>-λ_{min}（min}（a）$。我们表明，存在$ \ mathbb {f}^n $的子空间$ \ mathcal {y} $，独立于$ b $，使得$ \ {x_ {x_ {b，ω} - x__ {b，μ} \midΩ $ \ dim（\ Mathcal {y}）\ leq \ dim（\ Mathcal {s} + a \ Mathcal {s}） - \ dim（\ dimcal {s}）= \ shatbf {indbf {ind} _a（\ nathcal {s}）尊重$ a $。特别是，如果$ \ Mathcal {s} $是Krylov子空间，这意味着Hallman＆Gu（2018）的低维度结果。问题也是如此，当$ a $为正时，$ \ natercal {s} $是Krylov子空间，它以$ω= 0 $的价格减少到CG，而minres则以$ω\ to \ infty $减少。我们研究了与$ a $ a $ and $ \ mathcal {s} $相关的$ \ mathbf {ind} _a（\ mathcal {s}）$的几个属性。我们表明，包含解决方案$ x_ {b，ω} $的仿射子空间$ \ mathcal {x} _b $的尺寸可能小于$ \ mathbf {ind} _a（\ mathcal {s}）$的所有$ b $。但是，我们还对$ a $ a和$ \ mathcal {s} $表现出足够的条件，在此下，$ \ mathcal {x}：= \ text {span} {\ {x_ {x_ {x_ {b，ω} - x_____________________ {b，μ} -λ_{min}（a）\}} $具有等于$ \ mathbf {ind} _a（\ Mathcal {s}）$的尺寸。然后，我们研究地图$ω\ mapsto x_ {b，ω} $的注射率，导致我们获得了Hallman＆Gu（2018）的凸性证明。我们结束表明该集合，例如$ m（\ nathcal {s}，\ Mathcal {s}'）= \ {a \ in \ Mathbb {f}^{n \ times n} \ times n} \ mid \ mid \ mathcal {s} $ \ MATHCAL {S} \ subseteq \ Mathcal {s}'\ subseteq \ Mathbb {f}^n $，形成平滑的真实歧管，并探索它们之间的一些拓扑关系。
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.
