Generalized Forward-Backward Splitting

This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$'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 $n = 1$ non-smooth function, our method generalizes it to the case of arbitrary $n$. Our method makes an explicit use of the regularity of $F$ in the forward step, and the proximity operators of the $G_i$'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 $F$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.Comment: 24 pages, 4 figure

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 $J$ and $G$ 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

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 $\ell^2$-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

Howling on the edge: Mantled howler monkey (Alouatta palliata) howling behaviour and anthropogenic edge effects in a fragmented tropical rainforest in Costa Rica

The function of long calling is a subject of interest across animal behaviour study, particularly within primatology. Many primate species have male‐specific long‐distance calls, including platyrrhines like the folivorous howler monkey (Alouatta spp.). Howler monkeys may howl to defend resources such as feeding trees or areas of rich vegetation from other monkey groups. This study tests the ecological resource defence hypothesis for howling behaviour in the mantled howler monkey (Alouatta palliata) and investigates how anthropogenic forest fragmentation may influence howling behaviour. More specifically, this study examines how howling bout rate, duration, precursors and tree species richness, DBH, and canopy cover vary in 100 m anthropogenic edge and interior forest zones at La Suerte Biological Research Station (LSBRS), a fragmented tropical rainforest in Costa Rica. Results show that tree species richness and canopy cover are higher in forest interior at this site, suggesting that monkeys should howl at greater rates in the interior to defend access to these higher‐quality vegetation resources. Overall, our results supported the ecological resource defence hypothesis. The main howl precursor was howling from neighbouring groups. Although howling rate did not differ between forest zones, howling bouts from forest interior were longer, had a greater number of howls per bout and were preceded by different precursors than howls from anthropogenic edge zones, including more howls from neighbouring groups. Our findings provide some of the first evidence for behavioural edge effects in primate vocal communication behaviour