arXiv subjects

Samuel Vaiter

Publications and source records attributed to Samuel Vaiter.

57 records · Page 4Linked to original sources

arXiv2012

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.

math.ST↗

arXiv2012

Robust Sparse Analysis Regularization

This paper investigates the theoretical guarantees of L1-analysis regularization when solving linear inverse problems. Most of previous works in the literature have mainly focused on the sparse synthesis prior where the sparsity is measured as the L1 norm of the coefficients that synthesize the signal from a given dictionary. In contrast, the more general analysis regularization minimizes the L1 norm of the correlations between the signal and the atoms in the dictionary, where these correlations define the analysis support. The corresponding variational problem encompasses several well-known regularizations such as the discrete total variation and the Fused Lasso. Our main contributions consist in deriving sufficient conditions that guarantee exact or partial analysis support recovery of the true signal in presence of noise. More precisely, we give a sufficient condition to ensure that a signal is the unique solution of the L1-analysis regularization in the noiseless case. The same condition also guarantees exact analysis support recovery and L2-robustness of the L1-analysis minimizer vis-a-vis an enough small noise in the measurements. This condition turns to be sharp for the robustness of the analysis support. To show partial support recovery and L2-robustness to an arbitrary bounded noise, we introduce a stronger sufficient condition. When specialized to the L1-synthesis regularization, our results recover some corresponding recovery and robustness guarantees previously known in the literature. From this perspective, our work is a generalization of these results. We finally illustrate these theoretical findings on several examples to study the robustness of the 1-D total variation and Fused Lasso regularizations.

cs.IT↗

arXiv2012

The Degrees of Freedom of the Group Lasso

This paper studies the sensitivity to the observations of the block/group Lasso solution to an overdetermined linear regression model. Such a regularization is known to promote sparsity patterns structured as nonoverlapping groups of coefficients. Our main contribution provides a local parameterization of the solution with respect to the observations. As a byproduct, we give an unbiased estimate of the degrees of freedom of the group Lasso. Among other applications of such results, one can choose in a principled and objective way the regularization parameter of the Lasso through model selection criteria.

math.ST↗