95 research outputs found
Existence and approximation of fixed points of right Bregman nonexpansive operators
We study the existence and approximation of fixed points of right Bregman nonexpansive operators in reflexive Banach space. We present, in particular, necessary and sufficient conditions for the existence of fixed points and an implicit scheme for approximating them
Convergence and Perturbation Resilience of Dynamic String-Averaging Projection Methods
We consider the convex feasibility problem (CFP) in Hilbert space and
concentrate on the study of string-averaging projection (SAP) methods for the
CFP, analyzing their convergence and their perturbation resilience. In the
past, SAP methods were formulated with a single predetermined set of strings
and a single predetermined set of weights. Here we extend the scope of the
family of SAP methods to allow iteration-index-dependent variable strings and
weights and term such methods dynamic string-averaging projection (DSAP)
methods. The bounded perturbation resilience of DSAP methods is relevant and
important for their possible use in the framework of the recently developed
superiorization heuristic methodology for constrained minimization problems.Comment: Computational Optimization and Applications, accepted for publicatio
Global convergence of a non-convex Douglas-Rachford iteration
We establish a region of convergence for the proto-typical non-convex
Douglas-Rachford iteration which finds a point on the intersection of a line
and a circle. Previous work on the non-convex iteration [2] was only able to
establish local convergence, and was ineffective in that no explicit region of
convergence could be given
New results on q-positivity
In this paper we discuss symmetrically self-dual spaces, which are simply
real vector spaces with a symmetric bilinear form. Certain subsets of the space
will be called q-positive, where q is the quadratic form induced by the
original bilinear form. The notion of q-positivity generalizes the classical
notion of the monotonicity of a subset of a product of a Banach space and its
dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss
concepts generalizing the representations of monotone sets by convex functions,
as well as the number of maximally q-positive extensions of a q-positive set.
We also discuss symmetrically self-dual Banach spaces, in which we add a Banach
space structure, giving new characterizations of maximal q-positivity. The
paper finishes with two new examples.Comment: 18 page
Incremental proximal methods for large scale convex optimization
Laboratory for Information and Decision Systems Report LIDS-P-2847We consider the minimization of a sum∑m [over]i=1 fi (x) consisting of a large
number of convex component functions fi . For this problem, incremental methods
consisting of gradient or subgradient iterations applied to single components have
proved very effective. We propose new incremental methods, consisting of proximal
iterations applied to single components, as well as combinations of gradient, subgradient,
and proximal iterations. We provide a convergence and rate of convergence
analysis of a variety of such methods, including some that involve randomization in
the selection of components.We also discuss applications in a few contexts, including
signal processing and inference/machine learning.United States. Air Force Office of Scientific Research (grant FA9550-10-1-0412
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
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