No abstract

The aim of this paper is to introduce and study a two-step debiasing method for variational regularization. After solving the standard variational problem, the key idea is to add a consecutive debiasing step minimizing the data fidelity on an appropriate set, the so-called model manifold. The latter is defined by Bregman distances or infimal convolutions thereof, using the (uniquely defined) subgradient appearing in the optimality condition of the variational method. For particular settings, such as anisotropic 1 and TV-type regularization, previously used debiasing techniques are shown to be special cases. The proposed approach is however easily applicable to a wider range of regularizations. The two-step debiasing is shown to be well-defined and to optimally reduce bias in a certain setting.In addition to visual and PSNR-based evaluations, different notions of bias and variance decompositions are investigated in numerical studies. The improvements offered by the proposed scheme are demonstrated and its performance is shown to be comparable to optimal results obtained with Bregman iterations.where ∂ u H is the derivative of H with respect to the first argument. Now we proceed to a second step, where we only keep the subgradient p α and minimizêObviously, this problem is only of interest if there is no one-to-one relation between subgradients and primal values u, otherwise we always obtainû α = u α . The most interesting case with respect to applications is the one of J being absolutely one-homogeneous, i.e. J(λu) = |λ|J(u) for all λ ∈ R, where the subdifferential can be multivalued at least at u = 0.The debiasing step can be reformulated in an equivalent way aswith the (generalized) Bregman distance given by. We remark that for absolutely one-homogeneous J this simplifies toThe reformulation in terms of a Bregman distance indicates a first connection to Bregman iterations, which we make more precise in the sequel of the paper.Summing up, we examine the following twostep method: 1) Compute the (biased) solution u α of (1.1) with optimality condition (1.2),2) Compute the (debiased) solutionû α as the minimizer of (1.3) or equivalently (1.4).In order to relate further to the previous approaches of debiasing 1 -minimizers given only the support and not the sign, as well as the approach with linear model subspaces, we consider another debiasing approach being blind against the sign. The natural generalization in the case of an absolutely one-homogeneous functional J is to replace the second step bydenotes the infimal convolution between the Bregman distances D pα J (·, u α ) and D −pα J (·, −u α ), evaluated at u ∈ X .The infimal convolution of two functionals F and G on a Banach space X is defined asConsequently,û α is also a solution of (3.3). Theorem 4.12. The set M B = {u ∈ X | D p J (u, v) = 0} is a nonempty convex cone. Proof. The map u → D p J (u, v) is convex and nonnegative, hence {u | D p J (u, v) = 0} = {u | D p J (u, v) ≤ 0} is convex as a sublevel set of a convex functional. Moreover, for each c ≥ 0 we h...

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