Matches in SemOpenAlex for { <https://semopenalex.org/work/W3081440720> ?p ?o ?g. }
Showing items 1 to 76 of
76
with 100 items per page.
- W3081440720 abstract "We study the problem $x_{b,omega} := text{arg min}_{x in mathcal{S}} |(A + omega I)^{-1/2} (b - Ax)|_2$, with $A = A^*$, for a subspace $mathcal{S}$ of $mathbb{F}^n$ ($mathbb{F} = mathbb{R}$ or $mathbb{C}$), and $omega > -lambda_{min}(A)$. We show that there exists a subspace $mathcal{Y}$ of $mathbb{F}^n$, independent of $b$, such that ${x_{b,omega} - x_{b,mu} mid omega,mu > -lambda_{min}(A)} subseteq mathcal{Y}$, where $dim(mathcal{Y}) leq dim(mathcal{S} + Amathcal{S}) - dim(mathcal{S}) = mathbf{Ind}_A(mathcal{S})$, 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 $omega = 0$ and to MINRES for $omega to infty$. 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 $x_{b,omega}$ 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 $mathcal{X} := text{Span}{{x_{b,omega} - x_{b,mu} mid b in mathbb{F}^n, omega,mu > -lambda_{min}(A)}}$ has dimension equal to $mathbf{Ind}_A(mathcal{S})$. We then study the injectivity of the map $omega mapsto x_{b,omega}$, leading us to a proof of the convexity result from Hallman & Gu (2018). We finish by showing that sets such as $M(mathcal{S},mathcal{S}') = {A in mathbb{F}^{n times n} mid mathcal{S} + Amathcal{S} = mathcal{S}'}$, for nested subspaces $mathcal{S} subseteq mathcal{S}' subseteq mathbb{F}^n$, form smooth real manifolds, and explore some topological relationships between them." @default.
- W3081440720 created "2020-09-01" @default.
- W3081440720 creator A5089335913 @default.
- W3081440720 creator A5090210790 @default.
- W3081440720 date "2020-08-25" @default.
- W3081440720 modified "2023-09-27" @default.
- W3081440720 title "The Index of Invariance and its Implications for a Parameterized Least Squares Problem." @default.
- W3081440720 cites W1557324374 @default.
- W3081440720 cites W1557899381 @default.
- W3081440720 cites W1973516374 @default.
- W3081440720 cites W1974511160 @default.
- W3081440720 cites W1986951886 @default.
- W3081440720 cites W2015469407 @default.
- W3081440720 cites W2015700462 @default.
- W3081440720 cites W2018850201 @default.
- W3081440720 cites W2040425799 @default.
- W3081440720 cites W2065205300 @default.
- W3081440720 cites W2097897435 @default.
- W3081440720 cites W2316564661 @default.
- W3081440720 cites W2885042145 @default.
- W3081440720 cites W2962892241 @default.
- W3081440720 cites W3017463949 @default.
- W3081440720 hasPublicationYear "2020" @default.
- W3081440720 type Work @default.
- W3081440720 sameAs 3081440720 @default.
- W3081440720 citedByCount "0" @default.
- W3081440720 crossrefType "posted-content" @default.
- W3081440720 hasAuthorship W3081440720A5089335913 @default.
- W3081440720 hasAuthorship W3081440720A5090210790 @default.
- W3081440720 hasConcept C114614502 @default.
- W3081440720 hasConcept C121332964 @default.
- W3081440720 hasConcept C12362212 @default.
- W3081440720 hasConcept C165464430 @default.
- W3081440720 hasConcept C202444582 @default.
- W3081440720 hasConcept C2778113609 @default.
- W3081440720 hasConcept C2779557605 @default.
- W3081440720 hasConcept C33676613 @default.
- W3081440720 hasConcept C33923547 @default.
- W3081440720 hasConcept C62520636 @default.
- W3081440720 hasConceptScore W3081440720C114614502 @default.
- W3081440720 hasConceptScore W3081440720C121332964 @default.
- W3081440720 hasConceptScore W3081440720C12362212 @default.
- W3081440720 hasConceptScore W3081440720C165464430 @default.
- W3081440720 hasConceptScore W3081440720C202444582 @default.
- W3081440720 hasConceptScore W3081440720C2778113609 @default.
- W3081440720 hasConceptScore W3081440720C2779557605 @default.
- W3081440720 hasConceptScore W3081440720C33676613 @default.
- W3081440720 hasConceptScore W3081440720C33923547 @default.
- W3081440720 hasConceptScore W3081440720C62520636 @default.
- W3081440720 hasLocation W30814407201 @default.
- W3081440720 hasOpenAccess W3081440720 @default.
- W3081440720 hasPrimaryLocation W30814407201 @default.
- W3081440720 hasRelatedWork W1984478706 @default.
- W3081440720 hasRelatedWork W2262047358 @default.
- W3081440720 hasRelatedWork W2346033033 @default.
- W3081440720 hasRelatedWork W2429100642 @default.
- W3081440720 hasRelatedWork W2748792156 @default.
- W3081440720 hasRelatedWork W2771153333 @default.
- W3081440720 hasRelatedWork W2804134287 @default.
- W3081440720 hasRelatedWork W2905744841 @default.
- W3081440720 hasRelatedWork W2950200016 @default.
- W3081440720 hasRelatedWork W2952343876 @default.
- W3081440720 hasRelatedWork W2953060771 @default.
- W3081440720 hasRelatedWork W2969352712 @default.
- W3081440720 hasRelatedWork W3005185859 @default.
- W3081440720 hasRelatedWork W3031266158 @default.
- W3081440720 hasRelatedWork W3083746203 @default.
- W3081440720 hasRelatedWork W3098184366 @default.
- W3081440720 hasRelatedWork W3099628207 @default.
- W3081440720 hasRelatedWork W3106298863 @default.
- W3081440720 hasRelatedWork W3107723504 @default.
- W3081440720 hasRelatedWork W3199762456 @default.
- W3081440720 isParatext "false" @default.
- W3081440720 isRetracted "false" @default.
- W3081440720 magId "3081440720" @default.
- W3081440720 workType "article" @default.