Abstract

Sparse recovery via L1-penalized minimization can be typically solved by iteratively reweighted least squares (IRLS). The reconstruction quality is generally sensitive to the value of regularization parameter. In this wok, we propose a novel data-driven optimization scheme based on minimization of unbiased predictive risk estimate (UPRE). For a given IRLS iterate, we decompose and linearize the update by matrix splitting strategy, which makes the computation of Jacobian matrix tractable; and the UPRE of the IRLS iterate can be recursively evaluated. The developed recursive UPRE enables us to monitor the reconstruction error during the IRLS iterations, which can be applied to optimize regularization parameter. Numerical tests demonstrate the reliability of the recursive UPRE and the optimality of sparse reconstruction.

Highlights

  • IntroductionConsider the following standard linear inverse problem: estimate the unknown coefficients model(1,2): from a linear (1)

  • Consider the following standard linear inverse problem: estimate the unknown coefficients model(1,2): from a linear (1)where is the observed data, is a deterministic design matrix, is a vector of i.i.d.zero-mean Gaussian random variable with known variance

  • For the non-linear sparse estimate by a given Iteratively reweighted least squares (IRLS) iterate, we develop a recursive procedure for computing unbiased predictive risk estimate (UPRE), which enables us to evaluate the prediction error for the sparse recovery

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Summary

Introduction

Consider the following standard linear inverse problem: estimate the unknown coefficients model(1,2): from a linear (1). Many regularized iterative reconstruction algorithms, e.g. IRLS(6) and FISTA(4), often require selection of the appropriate value for λ. Since EPE is inaccessible in practice due to unknown μ and x, the unbiased predictive risk estimate (UPRE). Is a Jacobian matrix defined as: Any estimate can be regarded as a linear or non-linear transformation of the observed data. . For the non-linear sparse estimate considered here, the computation of becomes essential for the UPRE evaluation. The UPRE has been widely used in the statistics and signal processing communities, as a principled and efficient way for parameter selection with a variety of linear and non-linear estimators(7,8,12,13). For the non-linear sparse estimate by a given IRLS iterate, we develop a recursive procedure for computing UPRE, which enables us to evaluate the (estimated) prediction error for the sparse recovery.

Matrix splitting to solve IRLS
Basic scheme of IRLS
Recursion of Jacobian matrix
Short summary
Random example
Deconvolution of 1-D signal
Conclusions
Full Text
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