Method of Lagrange Multipliers for Normalized Zero Norm Minimization

Bamrung Tausiesakul,Alessandro Formisano

doi:10.1155/2022/8711843

Bamrung Tausiesakul, Alessandro Formisano

Open Access

https://doi.org/10.1155/2022/8711843

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Abstract

We present a normalization of the p -norm. A compressive sensing criterion is proposed using the normalized zero norm. Based on the method of Lagrange multipliers, we derive the solution of the proposed optimization framework. It turns out that the new solution is a limit case of the least fractional norm solution for p = 0 , where its fixed-point iteration algorithm can readily follow an existing algorithm. The derivation of the minimal normalized zero norm solution herein gives a relation in the aspect of Lagrange multiplier method to existing works that invoke least fractional norm and least pseudo zero norm criteria.

Highlights

Various applications in science and engineering need to recover a desired signal x ∈ RN×1 from a set of observed data or measured data b ∈ RM×1 based on a modeling or measurement matrix A ∈ RM×N, which either depends on the model or can be chosen beforehand, for N ∈ N1×1 and
E signal can be recovered by solving an optimization problem related to linear least squares (LLS), i.e., x􏽢LLS
S.t. x ∈ ⎪⎩ 􏽮x ∈ RN×1|‖Ax − b‖2 < ε􏽯, noisy, where ε is the square root of the maximal allowable noise power [1]

Summary

Introduction

Various applications in science and engineering need to recover a desired signal x ∈ RN×1 from a set of observed data or measured data b ∈ RM×1 based on a modeling or measurement matrix A ∈ RM×N, which either depends on the model or can be chosen beforehand, for N ∈ N1×1 and. E signal can be recovered by solving an optimization problem related to linear least squares (LLS), i.e., x􏽢LLS. The l0-norm is originally adopted to impose the zero elements in the solution. By using the method of Lagrange multipliers, the proposed constrained optimization is solved and the emerging solution is equal to the limit case of that given by the least fractional norm for p 0. E solution is found in a closed form and turns to be a limiting case to that of the least fractional norm criterion in the former works.

Conventional Zero Norm

Normalized Zero Norm

Compressive Sensing

Compressive Sensing by Normalized Zero Norm

Numerical Examples

Conclusion