On ℓ 1 -Regularization in Light of Nashed's Ill-Posedness Concept

Journal of Inverse and Ill-Posed Problems

Plato

2017

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We consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces.It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case.Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.

Section: Introductionmentioning

confidence: 99%

On ill-posedness concepts, stable solvability and saturation

Journal of Inverse and Ill-Posed Problems

Plato

2017

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“…provides us with a variational inequality (13). Along the lines of the proof of [5, Theorem 5.2] one can show the assertion of the following lemma.…”

Section: Convergence Rates For 1 -Regularizationmentioning

confidence: 89%

“…In other words, for such operators there exist appropriate sequences {γn} n∈N occurring in (17) such that a variational source condition (13) holds for an index function ϕ from (17) and constant β = 1−µ 1+µ (see Proposition 5.5 below). Item (b) in Property 4.2 is a generalization of (9).…”

Section: (I)mentioning

confidence: 99%

“…With the paper [5] as starting point and preferably based on variational source conditions first introduced in [23], convergence rates for 1 -regularization of operator equations (1) and variants like elastic-net 1 p Ax − y δ p Y + α 1 2 x 2 + η x 1 → min, subject to x ∈ 1 (4) have been verified under the condition that the sparsity assumption slightly fails (cf. [6,13,14]). This means that the solution x † ∈ 1 is not sparse, abbreviated as x † / ∈ 0 .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On ℓ 1 -Regularization Under Continuity of the Forward Operator in Weaker Topologies

Gerth

2018

Trends in Mathematics

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Our focus is on the stable approximate solution of linear operator equations based on noisy data by using 1 -regularization as a sparsity-enforcing version of Tikhonov regularization. We summarize recent results on situations where the sparsity of the solution slightly fails. In particular, we show how the recently established theory for weak*-to-weak continuous linear forward operators can be extended to the case of weak*-to-weak* continuity. This might be of interest when the image space is non-reflexive. We discuss existence, stability and convergence of regularized solutions. For injective operators, we will formulate convergence rates by exploiting variational source conditions. The typical rate function obtained under an ill-posed operator is strictly concave and the degree of failure of the solution sparsity has an impact on its behavior. Linear convergence rates just occur in the two borderline cases of proper sparsity, where the solutions belong to 0 , and of well-posedness. For an exemplary operator, we demonstrate that the technical properties used in our theory can be verified in practice. In the last section, we briefly mention the difficult case of oversmoothing regularization where x † does not belong to 1 .

“…Since the synthesis operator L : ℓ 1 (N) → X is injective and bounded, the linear operator A : ℓ 1 (N) → Y is also injective and bounded if A : X → Y is. In [12,Proposition 4.6] we have shown that the operator equation (1.1) with A = A • L as introduced above is always ill-posed, i.e. R(A) = R(A), and even ill-posed of type II in the sense of Nashed (cf.…”

Section: Introductionmentioning

confidence: 99%

A unified approach to convergence rates for ℓ¹-regularization and lacking sparsity

Flemming

Journal of Inverse and Ill-Posed Problems

Veselić

2015

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In ℓ 1 -regularization, which is an important tool in signal and image processing, one usually is concerned with signals and images having a sparse representation in some suitable basis, e.g. in a wavelet basis. Many results on convergence and convergence rates of sparse approximate solutions to linear ill-posed problems are known, but rate results for the ℓ 1 -regularization in case of lacking sparsity had not been published until 2013. In the last two years, however, two articles appeared providing sufficient conditions for convergence rates in case of nonsparse but almost sparse solutions. In the present paper, we suggest a third sufficient condition, which unifies the existing two and, by the way, also incorporates the well-known restricted isometry property. MSC2010 subject classification: 65J20, 47A52, 49N45