Minimization of the logarithmic function in sparse recovery

Abstract In sparse information recovery, the key issue is to solve the l0-minimization which is NP-hard. Therefore we consider the logarithmic alternative function to replace 0-norm. In this paper, theoretical analysis of logarithmic alternative model is given in detail. First, we discuss the equivalence between l0-minimization and the logarithmic function minimization. It is proved that the solution of the alternative model also solves the l0-minimization in both noiseless and noisy cases. Second, the recovery condition of logarithmic minimization is presented. By this new condition and a new concept named Logarithms Null Space Constant (L-NSC), we demonstrate the advantages of this model over classical approaches. Third, we study the optimal properties of the alternative model including the analytic property of optimal solution, a fixed point iterative algorithm and the corresponding convergence conclusion. Finally, we use the logarithmic function and the proposed algorithm to solve the multi-source time difference of arrival (TDOA) location problem. Compared with classic approaches, the experimental results demonstrate the effectiveness of the proposed algorithm.

[1]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[2]  Jia Liang,et al.  A Digital False-Target Image Synthesizer Method Against ISAR Based on Polyphase Code and Sub-Nyquist Sampling , 2020, IEEE Geoscience and Remote Sensing Letters.

[3]  Wenhua Hu,et al.  A Bistatic ISAR Sparse Aperture High Resolution Imaging Algorithm based on ROMP Algorithm , 2019, 2019 IEEE 3rd Information Technology, Networking, Electronic and Automation Control Conference (ITNEC).

[4]  Cheng Ping,et al.  New Compressed Sensing ISAR Imaging Algorithm Based on Log–Sum Minimization , 2016 .

[5]  Yi Shen,et al.  Sparsity and incoherence in orthogonal matching pursuit , 2019, Multidimens. Syst. Signal Process..

[6]  Jicong Fan,et al.  Sparse subspace clustering for data with missing entries and high-rank matrix completion , 2017, Neural Networks.

[7]  S. Foucart,et al.  Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0 , 2009 .

[8]  Wasim Q. Malik,et al.  Computationally Efficient Algorithms for Sparse, Dynamic Solutions to the EEG Source Localization Problem , 2018, IEEE Transactions on Biomedical Engineering.

[9]  Trac D. Tran,et al.  Stepwise Optimal Subspace Pursuit for Improving Sparse Recovery , 2011, IEEE Signal Processing Letters.

[10]  Trac D. Tran,et al.  Sparse Signal Recovery via Generalized Entropy Functions Minimization , 2017, IEEE Transactions on Signal Processing.

[11]  K. C. Ho,et al.  A Study on the Effects of Sensor Position Error and the Placement of Calibration Emitter for Source Localization , 2014, IEEE Transactions on Wireless Communications.

[12]  Shijie Zhou,et al.  Localization of Activation Origin on Patient-Specific Endocardial Surface by the Equivalent Double Layer (EDL) Source Model With Sparse Bayesian Learning , 2019, IEEE Transactions on Biomedical Engineering.

[13]  Yu Zhijun,et al.  Multiple Target Localization via Compressed Sensing in Wireless Sensor Networks , 2012 .

[14]  Ely Porat,et al.  For-All Sparse Recovery in Near-Optimal Time , 2014, ACM Trans. Algorithms.

[15]  Ivor W. Tsang,et al.  Matching Pursuit LASSO Part I: Sparse Recovery Over Big Dictionary , 2015, IEEE Transactions on Signal Processing.

[16]  Anru Zhang,et al.  Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-Rank Matrices , 2013, IEEE Transactions on Information Theory.

[17]  Gitta Kutyniok,et al.  PROMP: A sparse recovery approach to lattice-valued signals , 2017, Applied and Computational Harmonic Analysis.

[18]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[19]  Deyu Meng,et al.  Robust Low-Rank Matrix Factorization Under General Mixture Noise Distributions. , 2016 .

[20]  Qian Du,et al.  Equivalent-Sparse Unmixing Through Spatial and Spectral Constrained Endmember Selection From an Image-Derived Spectral Library , 2015, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing.

[21]  Ling-Hua Chang,et al.  An improved RIP-based performance guarantee for sparse signal reconstruction via subspace pursuit , 2014, 2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM).

[22]  Rick Chartrand,et al.  Exact Reconstruction of Sparse Signals via Nonconvex Minimization , 2007, IEEE Signal Processing Letters.

[23]  Farrokh Marvasti,et al.  Iterative Null-space Projection Method with Adaptive Thresholding in Sparse Signal Recovery and Matrix Completion , 2016, IET Signal Process..

[24]  Lei Zhang,et al.  High-Resolution ISAR Imaging and Motion Compensation With 2-D Joint Sparse Reconstruction , 2020, IEEE Transactions on Geoscience and Remote Sensing.

[25]  Prabhu Babu,et al.  Efficient Algorithms on Robust Low-Rank Matrix Completion Against Outliers , 2016, IEEE Transactions on Signal Processing.

[26]  Geert Leus,et al.  Sparsity-Aware Multi-Source TDOA Localization , 2013, IEEE Transactions on Signal Processing.

[27]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[28]  Yi Gao,et al.  Stability and robustness of the l2/lq-minimization for block sparse recovery , 2017, Signal Process..

[29]  Jigen Peng,et al.  $NP/CMP$ Equivalence: A Phenomenon Hidden Among Sparsity Models $l_{0}$ Minimization and $l_{p}$ , 2015, IEEE Transactions on Information Theory.

[30]  Jigen Peng,et al.  Minimization of Fraction Function Penalty in Compressed Sensing , 2017, IEEE Transactions on Neural Networks and Learning Systems.