We study the least squares problem x_{b,\omega} := \min _{x \in S} \|(A + \omega I)^{-1/2} (b - Ax)\|_2, with A = A^*, for a subspace S of F^n (F = R or C), and \omega > -min(A). We show that there exists a subspace Y of F^n, independent of b, such that \{x_{b,\omega} - x_{b,\mu} \mid \omega,\mu > -min(A)\} \subseteq Y, where \dim(Y) \leq \dim(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). \to  We study several properties of \mathbf{Ind}_A(\mathcal{S}) in relation to A and \mathcal{S}. We show that in general, 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 a related set \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} + 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.
We study the problem x_{b,\omega} := \text{arg min _{x \in S} \|(A + \omega I)^{-1/2} (b - Ax)\|_2, with A = A^*, for a subspace S of F^n (F = R or C), and \omega > -min(A). We show that there exists a subspace Y of F^n, independent of b, such that \{x_{b,\omega} - x_{b,\mu} \mid \omega,\mu > -min(A)\} \subseteq Y, where \dim(Y) \leq \dim(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 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} + 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.
D
least squares
R
\min
\text{arg min
A
The problem is also such that when A is positive and S is a Krylov subspace, it reduces to CG for \omega = 0 and to MINRES for \omega
A
\infty.
D
in general,
D
a related set