Convergence analysis in convex regularization depending on the smoothness degree of the penalizer
Abstract
The problem of minimization of the least squares functional with a smooth, lower semi-continuous, convex regularizer J(·) is considered to be solved. Over some compact and convex subset Ω of the Hilbert space H, the regularizer is implicitly defined as J(·) : Ck(Ω, H) → R+ where k ∈ \1,2\. So the cost functional associated with some given linear, compact and injective forward operator T :Ω⊂ H → H, align Fα(· , fδ) := 12 T( · ) - fδH2 + αJ(·) , align where fδ is the given perturbed data with its perturbation amount δ in it. Convergence of the regularized optimum solution φα(δ) ∈ argmin Fα(φ, fδ) to the true solution φ is analysed depending on the smoothness degree of the regularizer, i.e. the cases k ∈ \1,2\ in J(·) : Ck(Ω, H) → R+. In both cases, we define such a regularization parameter that is in cooperation with the condition align α(δ, fδ) ∈ \ α> 0 φαδ - fδ ≤ τδ\ , align for some fixed τ≥ 1. In the case of k = 2, we are able to evaluate the discrepancy φα(δ) - fδ≤ τδ with the Hessian Lipschitz constant LH of the functional Fα(· , fδ).
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