Haut PDF Linear inverse problems with noise: primal and primal-dual splitting

Linear inverse problems with noise: primal and primal-dual splitting

Linear inverse problems with noise: primal and primal-dual splitting

(α) t∈N converges to a (non-strict) global minimizer (P f 1 ,γ,ψ ) at the rate O(1/t) on the restricted duality gap. 3.3 Discussion Algorithm 1 and 2 share some similarities, but exhibit also important differences. For instance, the primal-dual algo- rithm enjoys a convergence rate that is not known for the primal algorithm. Furthermore, the latter necessitates two operator inversions that can only be done efficiently for some Φ and H, while the former involves only application of these linear operators and their adjoints. Consequently, Algo- rithm 2 can virtually handle any inverse problem with a bounded linear H. In case where the inverses can be done efficiently, e.g. deconvolution with a tight frame, both algo- rithms have comparable computational burden. In general, if other regularizations/constraints are imposed on the solu- tion, in the form of additional proper lsc convex terms that would appear in (P f 1 ,γ,ψ ), both algorithms still apply by
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Inverse problems with poisson noise: Primal and primal-dual splitting

Inverse problems with poisson noise: Primal and primal-dual splitting

In this paper, we propose a framework for solving lin- ear inverse problems when the observations are corrupted by Poisson noise. In order to form the data fidelity term, we take the exact Poisson likelihood. As a prior, the images to restore are assumed to be positive and sparsely represented in a dic- tionary of atoms. The solution to the inverse problem is cast as the minimization of a non-smooth convex functional, for which we prove well-posedness of the optimization problem, characterize the corresponding minimizers, and solve them by means of primal and primal-dual proximal splitting al- gorithms originating from the realm of non-smooth convex optimization theory. Convergence of the algorithms is also shown. Experimental results and comparison to other algo- rithms on deconvolution are finally conducted.
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A dual-primal coupling technique with local time step for wave propagation problems

A dual-primal coupling technique with local time step for wave propagation problems

must be determined. We point out that for each time intervale [t 2n , t 2n+2 ] there are three of these quantities. We must then write three linear independent equations that will allow us to obtain them and to couple both systems. The coupling equations: Energy conservation The additional equations that we will add in order to couple the two systems in (19) will be chosen in such a way that the stability of the scheme will be ensured a priori. A simple way to do that is to impose a discrete version of the energy conservation property explained on the remarks 2.1 and 4.1. We introduce the discrete energy at the even time steps by
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Interior-point methids of primal-dual central-path type for solving some classes of linear complementarity problemes over symmetric cones

Interior-point methids of primal-dual central-path type for solving some classes of linear complementarity problemes over symmetric cones

This thesis is concerned with the analysis, implementation of interior-point methods . In particular, we focus on two type of problems: horizontal linear complementarity prob- lems (HLCPs) and Semidefinite linear complementarity problems (SDLCPs). In chapter 1: we present the definitions and terms that will be used throughout the thesis. In chapter 2: we present a full-Newton feasible step interior-point algorithm for solving monotone horizontal linear complementarity problems. The idea of this algorithm is to follow the centers of the perturbed HLCP by using only full-Newton steps with the ad- vantage that no line search is required and restricts iterates in a small neighborhood of the central-path by introducing a suitable proximity measure during the solution process. Then we prove across a new appropriate choice of the defaults of the threshold of the parameter τ which defines the size of the neighborhood of the central-path and of the update barrier parameter θ that our algorithm is well-defined and the full-Newton step to the central-path is locally quadratically convergent. Moreover, we derive its complexity bound. which coincides with the best known iteration bound for such feasible IPMs. Fi- nally, we report some numerical results to show the ability of this approach.
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STOCHASTIC PRIMAL-DUAL HYBRID GRADIENT ALGORITHM WITH ARBITRARY SAMPLING AND IMAGING APPLICATIONS

STOCHASTIC PRIMAL-DUAL HYBRID GRADIENT ALGORITHM WITH ARBITRARY SAMPLING AND IMAGING APPLICATIONS

algorithm [ 37 , 21 , 13 , 36 , 14 , 15 ]. It has been used to solve a vast amount of state- of-the-art problems—to name a few examples in imaging: image denoising with the structure tensor [ 22 ], total generalized variation denoising [ 11 ], dynamic regularization [ 7 ], multi-modal medical imaging [ 27 ], multi-spectral medical imaging [ 43 ], computa- tion of non-linear eigenfunctions [ 26 ], regularization with directional total generalized variation [ 29 ]. Its popularity stems from two facts: First, it is very simple and there- fore easy to implement. Second, it involves only simple operations like matrix-vector multiplications and evaluations of proximal operators which are for many problems of interest simple and in closed-form or easy to compute iteratively, cf. e.g. [ 33 ]. However, for large problems that are encountered in many real world applications,
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Inverse Problems with Time-frequency Dictionaries and non-white Gaussian Noise

Inverse Problems with Time-frequency Dictionaries and non-white Gaussian Noise

Sparse regression consists in estimating unknown coefficients in a linear regression framework under the assumption that only a few regressors, also called variables or features, are predictive. In a word, most features are non-informative and their associated coefficients should be zero. The forward problem we consider in this work is the one proposed in [1] in the context of M/EEG. MEG and EEG are brain imaging methods that record the electromagnetic signals produced by active neurons using an array of sensors. The ambition is to use M/EEG to localize active brain regions non-invasively,
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A first-order primal-dual algorithm for convex problems with applications to imaging

A first-order primal-dual algorithm for convex problems with applications to imaging

1 Introduction Variational methods have proven to be particularly useful to solve a number of ill-posed inverse imaging problems. They can be divided into two fundamentally different classes: Convex and non-convex problems. The advantage of convex problems over non-convex problems is that a global optimum can be computed, in general with a good precision and in a reasonable time, independent of the initialization. Hence, the quality of the solution solely depends on the accuracy of the model. On the other hand, non-convex problems have often the ability to model more precisely the process behind an image acquisition, but have the drawback that the quality of the solution is more sensitive to the initialization and the optimization algorithm.
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A Class of Randomized Primal-Dual Algorithms for Distributed Optimization

A Class of Randomized Primal-Dual Algorithms for Distributed Optimization

1 Introduction There has been recently a growing interest in primal-dual approaches for finding a zero of a sum of monotone operators or minimizing a sum of proper lower-semicontinuous convex functions (see [ 36 ] and the references therein). When various linear operators are involved in the formulation of the problem under investigation, solving jointly its primal and dual forms allows the design of strategies where none of the linear operators needs to be inverted. Avoiding such inversions may offer a significant advantage in terms of computational complexity when dealing with large-scale problems (see e.g. [ 6 , 27 , 31 , 35 , 46 , 48 , 52 ]).
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Solving inverse problems with overcomplete transforms and convex optimization techniques

Solving inverse problems with overcomplete transforms and convex optimization techniques

Keywords: inverse problems, wavelets, redundant transforms, convex optimization, proximal operator, restoration 1. INTRODUCTION Because of sensor imperfection and acquisition mode, observed data are often noisy and degraded by a linear operator. This operator and noise properties may actually depend on the considered application. For in- stance, in optical satellite imaging, the linear operator models a blur and the noise can be assumed Gaussian. In the biomedical area, the ill-conditioned linear operator in parallel Magnetic Resonance Imaging (pMRI) represents a sensitivity matrix and the noise is Gaussian, whereas in Positron Emission Tomography (PET), the acquisition process is modelled by a large-size projection matrix and the noise is Poisson distributed.
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Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration

Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration

These methods require only approximations of the functions used in the formulation of the optimization problem, which is of the utmost importance for solving online signal processing problems. The almost sure convergence of these algorithms has been established. The stochastic version of the primal- dual algorithm that we have investigated has been evaluated in an online image restoration problem in which the data are blurred by a stochastic point spread function and corrupted with noise.

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Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators

Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators

1 Introduction Duality theory occupies a central place in classical optimization [ 19 , 24 , 33 , 40 , 41 ]. Since the mid 1960s it has expanded in various directions, e.g., variational inequalities [ 2 , 17 , 21 , 23 , 26 , 34 ], minimax and saddle point problems [ 27 , 29 , 32 , 39 ], and, from a more global perspective, monotone inclusions [ 5 , 9 , 10 , 16 , 31 , 37 , 38 ]. In the present paper, we propose an algorithm for solving the following structured duality framework for monotone inclusions that encompasses the above cited works. In this formulation, we denote by B  D the parallel sum of two set-valued operators B and D (see ( 2.5 )). This operation plays a central role in convex analysis and monotone operator theory. In particular, B  D can be seen as a regularization of B by D, and  is naturally connected to addition through duality since (B + D) −1 = B −1
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Dynamic Placement with Connectivity for RSNs based on a Primal-Dual Neural Network

Dynamic Placement with Connectivity for RSNs based on a Primal-Dual Neural Network

under constraints (2), (3), (4),(5), (6) and (8). 3 Neural network as a fast solver for linear quadratic programs The basic idea for solving an optimization problem using a tailored neural network is to make sure that the neural network will converge asymptotically at a fast rate and that the equilibrium point of the neural network will correspond effectively to the solution of the original optimization problem. In 1986, Tank and Hopfield introduced a linear programming neural network solver realized with an analogic circuit which appeared to be well suited for applications requiring on-line solutions [5]. After this first successful attempt, many neural network models for solving linear and quadratic programming problems have been proposed in the literature. For a review see [6,7] and an application see [10].
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The convex algebraic geometry of linear inverse problems

The convex algebraic geometry of linear inverse problems

These Positivstellensatz-based relaxations provide inner approximations to the set {q | kM † qk ∗ A ≤ 1}, and therefore a lower bound to the optimal value of the dual convex program (2). Consequently, they can be viewed as outer approxi- mations to the atomic norm ball C( A ). This interpretation leads to the following interesting conclusion: If a point ˜ x is an extreme point of the convex set C( A ) then the tangent cone at ˜ x with respect to C( A ) is smaller than the tangent cone at (an appropriately scaled) ˜ x with respect to the outer approximation. Thus we have a stronger requirement in terms of a larger number of measurements for exact recovery, which provides a trade-off between the complexity of the convex relaxation procedure and the amount of information required for exact recovery.
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A linear model approach for ultrasonic inverse problems with attenuation and dispersion

A linear model approach for ultrasonic inverse problems with attenuation and dispersion

The present paper aims at contributing to the solving of ultrasonic inverse problems by including attenuation and dispersion in the direct model. In particular, we propose to ac- count for physical attenuation profiles defined in the frequency domain such as power law attenuation models. Our objective is threefold. First, we improve the ultrasound model accuracy compared to the standard attenuation-free model. Second, in contrast with the methods described above [23–27], we yield a more constrained description of the data. Consequently, better performance of the inversion procedure is expected. In partic- ular, a more accurate model aims at improving echo detection for long propagation distances where the signal-to-noise ratio is low. Last, our framework yields a linear direct model which enables the use of many acknowledged inversion methods [8]. Related works [7, 28] proposed similar approaches but with an empirical description of attenuation within a time-domain signal model. The model that we propose is derived from the physics of wave propagation and is described in the frequency domain [3, 16].
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Local Linear Convergence Analysis of Primal-Dual Splitting Methods

Local Linear Convergence Analysis of Primal-Dual Splitting Methods

) [ 13 ] under strong convexity of either the primal or the dual problem. The iterates converge globally linearly if both the primal and dual problems are strongly convex [ 13 , 9 ], or locally linearly under certain regularity assumptions [ 35 ]. How- ever, in practice, local linear convergence of the sequences generated by Algorithm 1 has been observed for many problems with the absence of strong convexity (as confirmed by our numerical experiments in Section 6 ). None of the existing theoretical analysis was able to explain this behaviour so far. Providing the theoretical underpinnings of this local behaviour is the main goal pursued in this paper. Our main findings can be summarized as follows.
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Primal-dual interior point optimization for a regularized reconstruction of NMR relaxation time distributions

Primal-dual interior point optimization for a regularized reconstruction of NMR relaxation time distributions

[10] S. G. Nash, “A survey of truncated-Newton methods,” Journal of Computational and Applied Mathematics, vol. 124, pp. 45– 59, 2000. [11] E. Chouzenoux, S. Moussaoui, and J. Idier, “Majorize- minimize linesearch for inversion methods involving barrier function optimization,” Inverse Problems, vol. 28, no. 6, 2012. [12] A. Conn, N. Gould, and L. Toint, “A primal-dual algo- rithm for minimizing a nonconvex function subject to bounds and nonlinear constraints,” Tech. Rep. RC 20639, York- town Heights, NY, 1996, citeseer.ist.psu.edu/
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Analysis of decentralized potential field based multi-agent navigation via primal-dual Lyapunov theory

Analysis of decentralized potential field based multi-agent navigation via primal-dual Lyapunov theory

I. I NTRODUCTION The development of advanced decentralized collision avoidance algorithms for large scale systems is an issue of critical importance in future Air Traffic Control (ATC) architectures, where the increasing number of aircraft will render centralized approaches inefficient. In recent years the application of robotics collision avoidance potential field based methods to ATC has been explored [6],[15] as a promising alternative for such algorithms. A common prob- lem with potential field based path planning algorithms in multi-agent systems is the existence of local minima [7],[9]. The seminal work of Koditschek and Rimon [8] involved navigation of a single robot in an environment of spherical obstacles with guaranteed convergence. In previous work, the closed loop single robot navigation methodology of[8] was extended to multi-agent systems. In [10],[13] this method was extended to take into account the volume of each robot in a centralized multi-agent scheme, while a decentralized version has been presented in [4], [6]. Formation control for point agents using decentralized navigation functions was dealt with in [19], [3]. Decentralized navigation functions were also used for multiple UAV guidance in [2].
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Fast constrained least squares spectral unmixing using primal-dual interior point optimization

Fast constrained least squares spectral unmixing using primal-dual interior point optimization

bands from 383 nm to 2508 nm. A subset of P spectra is then randomly picked up to create synthetic mixtures with abundances simulated from a Dirichlet distribution. Only realizations with maximum abundance value lower than a specified level A max are retained. Finally, a random Gaussian noise is added to each resulting mixture spectrum, in order to get a signal to noise ratio (SNR) of 30 dB. The unmixing algorithms are implemented on Matlab 2012b and the calculations are performed using a HP Compaq Elite desktop having an Intel Core i7 3.4 GHz processor and 8 GB of RAM.
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Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradient

Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradient

Before going further, let us discuss how one quantifies convergence speed for saddle point problems. Several measures of optimality have been considered in the literature. The most natural one is feasibility error and optimality gap. It directly fits the definition of the optimization problem at stake. However, one cannot compute the optimality gap before the problem is solved. Hence, in algorithms, we usually use the Karush-Kuhn-Tucker (KKT) error instead. It is a computable quantity and if the Lagrangian’s gradient is metrically subregular [23], then a small KKT error implies that the current point is close to the set of saddle points. When the primal and dual domains are bounded, the duality gap is a very good way to measure optimality: it is often easily computable and it is an upper bound to the optimality gap. A generalization to unbounded domains has been proposed in [24]: the smoothed gap, based on the smoothing of nonsmooth functions [21], takes finite values for constrained problems, unlike the duality gap. Moreover, if the smoothness parameter is small and the smoothed gap is small, this means that optimality gap and feasibility error are both small. In the present paper, we shall reuse this concept not only for showing a convergence speed but also to define a new regularity assumption that we believe is better suited to the study of primal-dual algorithms.
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Aggregation for Linear Inverse Problems

Aggregation for Linear Inverse Problems

posedness, this term is of order n t−1 . Note that this term goes to zero when t is smaller than 1, yet hampering the consistency rate. In other cases and in the severely ill-posed setting, this term becomes dominant in the upper bound. Hence aggregation methods for inverse problems have the same kind of drawbacks than ℓ 1 penalization procedure since they cannot handle too badly ill-posed inverse problems.

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