281 research outputs found
Functions with Prescribed Best Linear Approximations
A common problem in applied mathematics is to find a function in a Hilbert
space with prescribed best approximations from a finite number of closed vector
subspaces. In the present paper we study the question of the existence of
solutions to such problems. A finite family of subspaces is said to satisfy the
\emph{Inverse Best Approximation Property (IBAP)} if there exists a point that
admits any selection of points from these subspaces as best approximations. We
provide various characterizations of the IBAP in terms of the geometry of the
subspaces. Connections between the IBAP and the linear convergence rate of the
periodic projection algorithm for solving the underlying affine feasibility
problem are also established. The results are applied to problems in harmonic
analysis, integral equations, signal theory, and wavelet frames
There is no variational characterization of the cycles in the method of periodic projections
The method of periodic projections consists in iterating projections onto
closed convex subsets of a Hilbert space according to a periodic sweeping
strategy. In the presence of sets, a long-standing question going
back to the 1960s is whether the limit cycles obtained by such a process can be
characterized as the minimizers of a certain functional. In this paper we
answer this question in the negative. Projection algorithms that minimize
smooth convex functions over a product of convex sets are also discussed
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
Asymptotic behavior of compositions of under-relaxed nonexpansive operators
In general there exists no relationship between the fixed point sets of the
composition and of the average of a family of nonexpansive operators in Hilbert
spaces. In this paper, we establish an asymptotic principle connecting the
cycles generated by under-relaxed compositions of nonexpansive operators to the
fixed points of the average of these operators. In the special case when the
operators are projectors onto closed convex sets, we prove a conjecture by De
Pierro which has so far been established only for projections onto affine
subspaces
HIPAD - A Hybrid Interior-Point Alternating Direction algorithm for knowledge-based SVM and feature selection
We consider classification tasks in the regime of scarce labeled training
data in high dimensional feature space, where specific expert knowledge is also
available. We propose a new hybrid optimization algorithm that solves the
elastic-net support vector machine (SVM) through an alternating direction
method of multipliers in the first phase, followed by an interior-point method
for the classical SVM in the second phase. Both SVM formulations are adapted to
knowledge incorporation. Our proposed algorithm addresses the challenges of
automatic feature selection, high optimization accuracy, and algorithmic
flexibility for taking advantage of prior knowledge. We demonstrate the
effectiveness and efficiency of our algorithm and compare it with existing
methods on a collection of synthetic and real-world data.Comment: Proceedings of 8th Learning and Intelligent OptimizatioN (LION8)
Conference, 201
Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm
The primal-dual optimization algorithm developed in Chambolle and Pock (CP),
2011 is applied to various convex optimization problems of interest in computed
tomography (CT) image reconstruction. This algorithm allows for rapid
prototyping of optimization problems for the purpose of designing iterative
image reconstruction algorithms for CT. The primal-dual algorithm is briefly
summarized in the article, and its potential for prototyping is demonstrated by
explicitly deriving CP algorithm instances for many optimization problems
relevant to CT. An example application modeling breast CT with low-intensity
X-ray illumination is presented.Comment: Resubmitted to Physics in Medicine and Biology. Text has been
modified according to referee comments, and typos in the equations have been
correcte
Compressed sensing imaging techniques for radio interferometry
Radio interferometry probes astrophysical signals through incomplete and
noisy Fourier measurements. The theory of compressed sensing demonstrates that
such measurements may actually suffice for accurate reconstruction of sparse or
compressible signals. We propose new generic imaging techniques based on convex
optimization for global minimization problems defined in this context. The
versatility of the framework notably allows introduction of specific prior
information on the signals, which offers the possibility of significant
improvements of reconstruction relative to the standard local matching pursuit
algorithm CLEAN used in radio astronomy. We illustrate the potential of the
approach by studying reconstruction performances on simulations of two
different kinds of signals observed with very generic interferometric
configurations. The first kind is an intensity field of compact astrophysical
objects. The second kind is the imprint of cosmic strings in the temperature
field of the cosmic microwave background radiation, of particular interest for
cosmology.Comment: 10 pages, 1 figure. Version 2 matches version accepted for
publication in MNRAS. Changes includes: writing corrections, clarifications
of arguments, figure update, and a new subsection 4.1 commenting on the exact
compliance of radio interferometric measurements with compressed sensin
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