40 research outputs found
Activity Identification and Local Linear Convergence of Douglas--Rachford/ADMM under Partial Smoothness
Convex optimization has become ubiquitous in most quantitative disciplines of
science, including variational image processing. Proximal splitting algorithms
are becoming popular to solve such structured convex optimization problems.
Within this class of algorithms, Douglas--Rachford (DR) and alternating
direction method of multipliers (ADMM) are designed to minimize the sum of two
proper lower semi-continuous convex functions whose proximity operators are
easy to compute. The goal of this work is to understand the local convergence
behaviour of DR (resp. ADMM) when the involved functions (resp. their
Legendre-Fenchel conjugates) are moreover partly smooth. More precisely, when
both of the two functions (resp. their conjugates) are partly smooth relative
to their respective manifolds, we show that DR (resp. ADMM) identifies these
manifolds in finite time. Moreover, when these manifolds are affine or linear,
we prove that DR/ADMM is locally linearly convergent. When and are
locally polyhedral, we show that the optimal convergence radius is given in
terms of the cosine of the Friedrichs angle between the tangent spaces of the
identified manifolds. This is illustrated by several concrete examples and
supported by numerical experiments.Comment: 17 pages, 1 figure, published in the proceedings of the Fifth
International Conference on Scale Space and Variational Methods in Computer
Visio
Generalized Forward-Backward Splitting
This paper introduces the generalized forward-backward splitting algorithm
for minimizing convex functions of the form , where
has a Lipschitz-continuous gradient and the 's are simple in the sense
that their Moreau proximity operators are easy to compute. While the
forward-backward algorithm cannot deal with more than non-smooth
function, our method generalizes it to the case of arbitrary . Our method
makes an explicit use of the regularity of in the forward step, and the
proximity operators of the 's are applied in parallel in the backward
step. This allows the generalized forward backward to efficiently address an
important class of convex problems. We prove its convergence in infinite
dimension, and its robustness to errors on the computation of the proximity
operators and of the gradient of . Examples on inverse problems in imaging
demonstrate the advantage of the proposed methods in comparison to other
splitting algorithms.Comment: 24 pages, 4 figure
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
HAE international home therapy consensus document
Hereditary angioedema (C1 inhibitor deficiency, HAE) is associated with intermittent swellings which are disabling and may be fatal. Effective treatments are available and these are most useful when given early in the course of the swelling. The requirement to attend a medical facility for parenteral treatment results in delays. Home therapy offers the possibility of earlier treatment and better symptom control, enabling patients to live more healthy, productive lives. This paper examines the evidence for patient-controlled home treatment of acute attacks ('self or assisted administration') and suggests a framework for patients and physicians interested in participating in home or self-administration programmes. It represents the opinion of the authors who have a wide range of expert experience in the management of HAE
Bayesian computation: a summary of the current state, and samples backwards and forwards
Multiple regulation of prolactin receptor gene expression in rat liver
International audienc
Segmentation of small animal PET/CT mouse brain scans using an MRI-based 3D digital atlas: application in a neuroinflammation mouse model.
The work reported in this paper aimed at developing and testing an automated method to calculate the biodistribution of a specific PET tracer in mouse brain PET/CT images using an MRI-based 3D digital atlas. Surface-based registration strategy and affine transformation estimation were considered. Such an approach allowed overcoming the lack of anatomical information in the inner regions of PET/CT brain scans. Promising results were obtained in one mouse (on two scans) and will be extended to a neuroinflammation mouse model to characterize the pathology and its evolution. Major improvements are expected regarding automation, time computation, robustness and reproducibility of mouse brain segmentation. Due to its generic implementation, this method could be successfully applied to PET/CT brain scans of other species (rat, primate) for which 3D digital atlases are available