The Index of Invariance and its Implications for a Parameterized Least Squares Problem
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We study the problem x_b,ω := arg min_x ∈𝒮(A + ω I)^-1/2 (b - Ax)_2, with A = A^*, for a subspace 𝒮 of 𝔽^n (𝔽 = ℝ or ℂ), and ω > -λ_min(A). We show that there exists a subspace 𝒴 of 𝔽^n, independent of b, such that {x_b,ω - x_b,μ|ω,μ > -λ_min(A)}⊆𝒴, where (𝒴) ≤(𝒮 + A𝒮) - (𝒮) = 𝐈𝐧𝐝_A(𝒮), a quantity which we call the index of invariance of 𝒮 with respect to A. In particular if 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 ω = 0 and to MINRES for ω→∞. We study several properties of 𝐈𝐧𝐝_A(𝒮) in relation to A and 𝒮. We show that the dimension of the affine subspace 𝒳_b containing the solutions x_b,ω can be smaller than 𝐈𝐧𝐝_A(𝒮) for all b. However, we also exhibit some sufficient conditions on A and 𝒮, under which 𝒳 := Span{x_b,ω - x_b,μ| b ∈𝔽^n, ω,μ > -λ_min(A)} has dimension equal to 𝐈𝐧𝐝_A(𝒮). We then study the injectivity of the map ω↦ 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(𝒮,𝒮') = {A ∈𝔽^n × n|𝒮 + A𝒮 = 𝒮'}, for nested subspaces 𝒮⊆𝒮' ⊆𝔽^n, form smooth real manifolds, and explore some topological relationships between them.
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