12,397 research outputs found
Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation
In this paper, we aim at recovering an unknown signal x0 from noisy
L1measurements y=Phi*x0+w, where Phi is an ill-conditioned or singular linear
operator and w accounts for some noise. To regularize such an ill-posed inverse
problem, we impose an analysis sparsity prior. More precisely, the recovery is
cast as a convex optimization program where the objective is the sum of a
quadratic data fidelity term and a regularization term formed of the L1-norm of
the correlations between the sought after signal and atoms in a given
(generally overcomplete) dictionary. The L1-sparsity analysis prior is weighted
by a regularization parameter lambda>0. In this paper, we prove that any
minimizers of this problem is a piecewise-affine function of the observations y
and the regularization parameter lambda. As a byproduct, we exploit these
properties to get an objectively guided choice of lambda. In particular, we
develop an extension of the Generalized Stein Unbiased Risk Estimator (GSURE)
and show that it is an unbiased and reliable estimator of an appropriately
defined risk. The latter encompasses special cases such as the prediction risk,
the projection risk and the estimation risk. We apply these risk estimators to
the special case of L1-sparsity analysis regularization. We also discuss
implementation issues and propose fast algorithms to solve the L1 analysis
minimization problem and to compute the associated GSURE. We finally illustrate
the applicability of our framework to parameter(s) selection on several imaging
problems
Geometric Duality for Convex Vector Optimization Problems
Geometric duality theory for multiple objective linear programming problems
turned out to be very useful for the development of efficient algorithms to
generate or approximate the whole set of nondominated points in the outcome
space. This article extends the geometric duality theory to convex vector
optimization problems.Comment: 21 page
Higher Weak Derivatives and Reflexive Algebras of Operators
Let D be a self-adjoint operator on a Hilbert space H and x a bounded
operator on H. We say that x is n-times weakly D-differentiable, if for any
pair of vectors a, b from H the function is n-times
differentiable. We give several characterizations of this property, among which
one is original. The results are used to show, that for a von Neumann algebra M
on H, the sub-algebra of n-times weakly D-differentiable operators has a
representation as a reflexive algebra of operators on a bigger Hilbert space.Comment: This version acknowledges results from the litterature, which the
first edition was unaware of. The result on the existence of a representation
with a reflexive image is ne
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