20 research outputs found
CLEAR: Covariant LEAst-square Re-fitting with applications to image restoration
In this paper, we propose a new framework to remove parts of the systematic
errors affecting popular restoration algorithms, with a special focus for image
processing tasks. Generalizing ideas that emerged for regularization,
we develop an approach re-fitting the results of standard methods towards the
input data. Total variation regularizations and non-local means are special
cases of interest. We identify important covariant information that should be
preserved by the re-fitting method, and emphasize the importance of preserving
the Jacobian (w.r.t. the observed signal) of the original estimator. Then, we
provide an approach that has a "twicing" flavor and allows re-fitting the
restored signal by adding back a local affine transformation of the residual
term. We illustrate the benefits of our method on numerical simulations for
image restoration tasks
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
Greedy-Like Algorithms for the Cosparse Analysis Model
The cosparse analysis model has been introduced recently as an interesting
alternative to the standard sparse synthesis approach. A prominent question
brought up by this new construction is the analysis pursuit problem -- the need
to find a signal belonging to this model, given a set of corrupted measurements
of it. Several pursuit methods have already been proposed based on
relaxation and a greedy approach. In this work we pursue this question further,
and propose a new family of pursuit algorithms for the cosparse analysis model,
mimicking the greedy-like methods -- compressive sampling matching pursuit
(CoSaMP), subspace pursuit (SP), iterative hard thresholding (IHT) and hard
thresholding pursuit (HTP). Assuming the availability of a near optimal
projection scheme that finds the nearest cosparse subspace to any vector, we
provide performance guarantees for these algorithms. Our theoretical study
relies on a restricted isometry property adapted to the context of the cosparse
analysis model. We explore empirically the performance of these algorithms by
adopting a plain thresholding projection, demonstrating their good performance
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
Local linear convergence of proximal coordinate descent algorithm
International audienceFor composite nonsmooth optimization problems, which are "regular enough", proximal gradient descent achieves model identification after a finite number of iterations. For instance, for the Lasso, this implies that the iterates of proximal gradient descent identify the non-zeros coefficients after a finite number of steps. The identification property has been shown for various optimization algorithms, such as accelerated gradient descent, Douglas-Rachford or variance-reduced algorithms, however, results concerning coordinate descent are scarcer. Identification properties often rely on the framework of "partial smoothness", which is a powerful but technical tool. In this work, we show that partial smooth functions have a simple characterization when the nonsmooth penalty is separable. In this simplified framework, we prove that cyclic coordinate descent achieves model identification in finite time, which leads to explicit local linear convergence rates for coordinate descent. Extensive experiments on various estimators and on real datasets demonstrate that these rates match well empirical results
Proximal Splitting Derivatives for Risk Estimation
This paper develops a novel framework to compute a projected Generalized Stein Unbiased Risk Estimator (GSURE) for a wide class of sparsely regularized solutions of inverse problems. This class includes arbitrary convex data fidelities with both analysis and synthesis mixed L1-L2 norms. The GSURE necessitates to compute the (weak) derivative of a solution w.r.t.~the observations. However, as the solution is not available in analytical form but rather through iterative schemes such as proximal splitting, we propose to iteratively compute the GSURE by differentiating the sequence of iterates. This provides us with a sequence of differential mappings, which, hopefully, converge to the desired derivative and allows to compute the GSURE. We illustrate this approach on total variation regularization with Gaussian noise and to sparse regularization with poisson noise, to automatically select the regularization parameter.Adaptivité pour la représentation des images naturelles et des texturesERC SIGMA-Visio