178 research outputs found
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
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
Convergence Rates with Inexact Non-expansive Operators
In this paper, we present a convergence rate analysis for the inexact
Krasnosel'skii-Mann iteration built from nonexpansive operators. Our results
include two main parts: we first establish global pointwise and ergodic
iteration-complexity bounds, and then, under a metric subregularity assumption,
we establish local linear convergence for the distance of the iterates to the
set of fixed points. The obtained iteration-complexity result can be applied to
analyze the convergence rate of various monotone operator splitting methods in
the literature, including the Forward-Backward, the Generalized
Forward-Backward, Douglas-Rachford, alternating direction method of multipliers
(ADMM) and Primal-Dual splitting methods. For these methods, we also develop
easily verifiable termination criteria for finding an approximate solution,
which can be seen as a generalization of the termination criterion for the
classical gradient descent method. We finally develop a parallel analysis for
the non-stationary Krasnosel'skii-Mann iteration. The usefulness of our results
is illustrated by applying them to a large class of structured monotone
inclusion and convex optimization problems. Experiments on some large scale
inverse problems in signal and image processing problems are shown.Comment: This is an extended version of the work presented in
http://arxiv.org/abs/1310.6636, and is accepted by the Mathematical
Programmin
Iteration-Complexity of a Generalized Forward Backward Splitting Algorithm
In this paper, we analyze the iteration-complexity of Generalized
Forward--Backward (GFB) splitting algorithm, as proposed in \cite{gfb2011}, for
minimizing a large class of composite objectives on a
Hilbert space, where has a Lipschitz-continuous gradient and the 's
are simple (\ie their proximity operators are easy to compute). We derive
iteration-complexity bounds (pointwise and ergodic) for the inexact version of
GFB to obtain an approximate solution based on an easily verifiable termination
criterion. Along the way, we prove complexity bounds for relaxed and inexact
fixed point iterations built from composition of nonexpansive averaged
operators. These results apply more generally to GFB when used to find a zero
of a sum of maximal monotone operators and a co-coercive operator on a
Hilbert space. The theoretical findings are exemplified with experiments on
video processing.Comment: 5 pages, 2 figure
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