arXiv subjects

Dirk A. Lorenz

Publications and source records attributed to Dirk A. Lorenz.

47 records · Page 3Linked to original sources

arXiv2011

Beyond convergence rates: Exact recovery with Tikhonov regularization with sparsity constraints

The Tikhonov regularization of linear ill-posed problems with an $\ell^1$ penalty is considered. We recall results for linear convergence rates and results on exact recovery of the support. Moreover, we derive conditions for exact support recovery which are especially applicable in the case of ill-posed problems, where other conditions, e.g. based on the so-called coherence or the restricted isometry property are usually not applicable. The obtained results also show that the regularized solutions do not only converge in the $\ell^1$-norm but also in the vector space $\ell^0$ (when considered as the strict inductive limit of the spaces $\R^n$ as $n$ tends to infinity). Additionally, the relations between different conditions for exact support recovery and linear convergence rates are investigated. With an imaging example from digital holography the applicability of the obtained results is illustrated, i.e. that one may check a priori if the experimental setup guarantees exact recovery with Tikhonov regularization with sparsity constraints.

math.FA↗

arXiv2011

Constructing test instances for Basis Pursuit Denoising

The number of available algorithms for the so-called Basis Pursuit Denoising problem (or the related LASSO-problem) is large and keeps growing. Similarly, the number of experiments to evaluate and compare these algorithms on different instances is growing. In this note, we present a method to produce instances with exact solutions which is based on a simple observation which is related to the so called source condition from sparse regularization.

cs.IT↗

arXiv2010

Morozov's principle for the augmented Lagrangian method applied to linear inverse problems

The Augmented Lagrangian Method as an approach for regularizing inverse problems received much attention recently, e.g. under the name Bregman iteration in imaging. This work shows convergence (rates) for this method when Morozov's discrepancy principle is chosen as a stopping rule. Moreover, error estimates for the involved sequence of subgradients are pointed out. The paper studies implications of these results for particular examples motivated by applications in imaging. These include the total variation regularization as well as $\ell^q$ penalties with $q\in[1,2]$. It is shown that Morozov's principle implies convergence (rates) for the iterates with respect to the metric of strict convergence and the $\ell^q$-norm, respectively.

math.NA↗

arXiv2010

Image sequence interpolation using optimal control

The problem of the generation of an intermediate image between two given images in an image sequence is considered. The problem is formulated as an optimal control problem governed by a transport equation. This approach bears similarities with the Horn \& Schunck method for optical flow calculation but in fact the model is quite different. The images are modelled in $BV$ and an analysis of solutions of transport equations with values in $BV$ is included. Moreover, the existence of optimal controls is proven and necessary conditions are derived. Finally, two algorithms are given and numerical results are compared with existing methods. The new method is competitive with state-of-the-art methods and even outperforms several existing methods.

cs.CV↗

arXiv2010

On Conditions for Convergence to Consensus

A new theorem on conditions for convergence to consensus of a multiagent time-dependent time-discrete dynamical system is presented. The theorem is build up on the notion of averaging maps. We compare this theorem to results by Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005) about set-valued Lyapunov theory and convergence under switching communication topologies. We give examples that point out differences of approaches including examples where Moreau's theorem is not applicable but ours is. Further on, we give examples that demonstrate that the theory of convergence to consensus is still not complete.

math.OC↗

arXiv2009

Error estimates for joint Tikhonov- and Lavrentiev-regularization of constrained control problems

We consider joint Tikhonov- and Lavrentiev-regularization of control problems with pointwise control- and state-constraints. We derive error estimates for the error which is introduced by the Tikhonov regularization. With the help of this results we show, that if the solution of the unconstrained problem has no active constraints, the same holds for the Tikhonov-regularized solution if the regularization parameter is small enough and a certain source condition is fulfilled.

math.OC↗

arXiv2009

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.

math.NA↗

arXiv2009

A projection proximal-point algorithm for l^1-minimization

The problem of the minimization of least squares functionals with $\ell^1$ penalties is considered in an infinite dimensional Hilbert space setting. While there are several algorithms available in the finite dimensional setting there are only a few of them which come with a proper convergence analysis in the infinite dimensional setting. In this work we provide an algorithm from a class which have not been considered for $\ell^1$ minimization before, namely a proximal-point method in combination with a projection step. We show that this idea gives a simple and easy to implement algorithm. We present experiments which indicate that the algorithm may perform better than other algorithms if we employ them without any special tricks. Hence, we may conclude that the projection proximal-point idea is a promising idea in the context of $\ell^1$-minimization.

math.NA↗

arXiv2008

Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

This paper addresses the regularization by sparsity constraints by means of weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main result it is proven that one gets a convergence rate in norm of $\sqrtδ$ for $1\leq p\leq 2$ as soon as the unknown solution is sparse. The case $p=1$ needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed. For $p<1$ only preliminary results are shown. These results indicate that, different from $p\geq 1$, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for $p=0$ shows that regularization need not to happen.

math.FA↗

arXiv2008

Linear convergence of iterative soft-thresholding

In this article a unified approach to iterative soft-thresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the so-called finite basis injectivity property or the minimizer possesses a so-called strict sparsity pattern. Moreover it is shown that the constants can be calculated explicitly in special cases (i.e. for compact operators). Furthermore, the techniques also can be used to establish linear convergence for related methods such as the iterative thresholding algorithm for joint sparsity and the accelerated gradient projection method.

math.FA↗

arXiv2008

A Semismooth Newton Method for Tikhonov Functionals with Sparsity Constraints

Minimization problems in $\ell^2$ for Tikhonov functionals with sparsity constraints are considered. Sparsity of the solution is ensured by a weighted $\ell^1$ penalty term. The necessary and sufficient condition for optimality is shown to be slantly differentiable (Newton differentiable), hence a semismooth Newton method is applicable. Local superlinear convergence of this method is proved. Numerical examples are provided which show that our method compares favorably with existing approaches.

math.OC↗