122 research outputs found
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
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
A constrained-based optimization approach for seismic data recovery problems
Random and structured noise both affect seismic data, hiding the reflections
of interest (primaries) that carry meaningful geophysical interpretation. When
the structured noise is composed of multiple reflections, its adaptive
cancellation is obtained through time-varying filtering, compensating
inaccuracies in given approximate templates. The under-determined problem can
then be formulated as a convex optimization one, providing estimates of both
filters and primaries. Within this framework, the criterion to be minimized
mainly consists of two parts: a data fidelity term and hard constraints
modeling a priori information. This formulation may avoid, or at least
facilitate, some parameter determination tasks, usually difficult to perform in
inverse problems. Not only classical constraints, such as sparsity, are
considered here, but also constraints expressed through hyperplanes, onto which
the projection is easy to compute. The latter constraints lead to improved
performance by further constraining the space of geophysically sound solutions.Comment: International Conference on Acoustics, Speech and Signal Processing
(ICASSP 2014); Special session "Seismic Signal Processing
A forward-backward view of some primal-dual optimization methods in image recovery
A wide array of image recovery problems can be abstracted into the problem of
minimizing a sum of composite convex functions in a Hilbert space. To solve
such problems, primal-dual proximal approaches have been developed which
provide efficient solutions to large-scale optimization problems. The objective
of this paper is to show that a number of existing algorithms can be derived
from a general form of the forward-backward algorithm applied in a suitable
product space. Our approach also allows us to develop useful extensions of
existing algorithms by introducing a variable metric. An illustration to image
restoration is provided
Stochastic Primal-Dual Three Operator Splitting with Arbitrary Sampling and Preconditioning
In this work we propose a stochastic primal-dual preconditioned
three-operator splitting algorithm for solving a class of convex
three-composite optimization problems. Our proposed scheme is a direct
three-operator splitting extension of the SPDHG algorithm [Chambolle et al.
2018]. We provide theoretical convergence analysis showing ergodic O(1/K)
convergence rate, and demonstrate the effectiveness of our approach in imaging
inverse problems
Multi-frequency image reconstruction for radio-interferometry with self-tuned regularization parameters
As the world's largest radio telescope, the Square Kilometer Array (SKA) will
provide radio interferometric data with unprecedented detail. Image
reconstruction algorithms for radio interferometry are challenged to scale well
with TeraByte image sizes never seen before. In this work, we investigate one
such 3D image reconstruction algorithm known as MUFFIN (MUlti-Frequency image
reconstruction For radio INterferometry). In particular, we focus on the
challenging task of automatically finding the optimal regularization parameter
values. In practice, finding the regularization parameters using classical grid
search is computationally intensive and nontrivial due to the lack of ground-
truth. We adopt a greedy strategy where, at each iteration, the optimal
parameters are found by minimizing the predicted Stein unbiased risk estimate
(PSURE). The proposed self-tuned version of MUFFIN involves parallel and
computationally efficient steps, and scales well with large- scale data.
Finally, numerical results on a 3D image are presented to showcase the
performance of the proposed approach
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