14 research outputs found
Optimal Convergence Rates for Tikhonov Regularization in Besov Scales
In this paper we deal with linear inverse problems and convergence rates for
Tikhonov regularization. We consider regularization in a scale of Banach
spaces, namely the scale of Besov spaces. We show that regularization in Banach
scales differs from regularization in Hilbert scales in the sense that it is
possible that stronger source conditions may lead to weaker convergence rates
and vive versa. Moreover, we present optimal source conditions for
regularization in Besov scales
Greedy Solution of Ill-Posed Problems: Error Bounds and Exact Inversion
The orthogonal matching pursuit (OMP) is an algorithm to solve sparse
approximation problems. Sufficient conditions for exact recovery are known with
and without noise. In this paper we investigate the applicability of the OMP
for the solution of ill-posed inverse problems in general and in particular for
two deconvolution examples from mass spectrometry and digital holography
respectively.
In sparse approximation problems one often has to deal with the problem of
redundancy of a dictionary, i.e. the atoms are not linearly independent.
However, one expects them to be approximatively orthogonal and this is
quantified by the so-called incoherence. This idea cannot be transfered to
ill-posed inverse problems since here the atoms are typically far from
orthogonal: The ill-posedness of the operator causes that the correlation of
two distinct atoms probably gets huge, i.e. that two atoms can look much alike.
Therefore one needs conditions which take the structure of the problem into
account and work without the concept of coherence. In this paper we develop
results for exact recovery of the support of noisy signals. In the two examples
in mass spectrometry and digital holography we show that our results lead to
practically relevant estimates such that one may check a priori if the
experimental setup guarantees exact deconvolution with OMP. Especially in the
example from digital holography our analysis may be regarded as a first step to
calculate the resolution power of droplet holography
Sparse Regularization with Penalty Term
We consider the stable approximation of sparse solutions to non-linear
operator equations by means of Tikhonov regularization with a subquadratic
penalty term. Imposing certain assumptions, which for a linear operator are
equivalent to the standard range condition, we derive the usual convergence
rate of the regularized solutions in dependence of the noise
level . Particular emphasis lies on the case, where the true solution
is known to have a sparse representation in a given basis. In this case, if the
differential of the operator satisfies a certain injectivity condition, we can
show that the actual convergence rate improves up to .Comment: 15 page
Sparsity and Compressed Sensing in Inverse Problems
This chapter is concerned with two important topics in the context of sparse recovery in inverse and ill-posed problems. In first part we elaborate condi-tions for exact recovery. In particular, we describe how both `1-minimization and matching pursuit methods can be used to regularize ill-posed problems and more-over, state conditions which guarantee exact recovery of the support in the sparse case. The focus of the second part is on the incomplete data scenario. We discuss ex-tensions of compressed sensing for specific infinite dimensional ill-posed measure-ment regimes. We are able to establish recovery error estimates when adequately relating the isometry constant of the sensing operator, the ill-posedness of the un-derlying model operator and the regularization parameter. Finally, we very briefly sketch how projected steepest descent iterations can be applied to retrieve the sparse solution
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem