Density Evolution Analysis of Node-Based Verification-Based Algorithms in Compressed Sensing
In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressed sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension n ). The asymptotic analysi...
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| Published in: | IEEE transactions on information theory Vol. 58; no. 10; pp. 6616 - 6645 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
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| Abstract | In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressed sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension n ). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration l in the asymptotic regime where n →∞. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of nontrivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of l . Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of n . Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is more accurate and simpler to implement. |
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| AbstractList | In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressed sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension ${n}$). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration ${ell}$ in the asymptotic regime where ${nrightarrowinfty}$. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of nontrivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of ${ell}$. Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of ${n}$ . Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is more accurate and simpler to - mplement. [PUBLICATION ABSTRACT] In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressed sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension n ). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration [ell] in the asymptotic regime where n arrow right infinity . The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of nontrivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of [ell] . Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of n . Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is more accurate and simpler to implement. In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressed sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension n ). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration l in the asymptotic regime where n →∞. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of nontrivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of l . Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of n . Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is more accurate and simpler to implement. |
| Author | Heidarzadeh, A. Eftekhari, Y. Lambadaris, I. Banihashemi, A. H. |
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| Keywords | Performance evaluation iterative recovery algorithms Input signal Differential equation Iterative method iterative decoding algorithms Iterative decoding Implementation low-density parity-check (LDPC) codes Sparse matrix Parity check codes Asymptotic analysis Deterministic approach verification-based recovery algorithms Large scale system Target tracking sparse graphs density evolution low-complexity compressed sensing Algorithm Channel coding success threshold message-passing algorithms Message passing Simulation Concentration effect Error correcting code sparse sensing matrix Compressed sensing |
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| SubjectTerms | Algorithm design and analysis Algorithms Applied sciences Asymptotic analysis Asymptotic methods Asymptotic properties channel coding Coding, codes Compressed sensing Data compression Decoding Density density evolution Detection Differential equations Exact sciences and technology Information theory Information, signal and communications theory iterative decoding algorithms iterative recovery algorithms low-complexity compressed sensing low-density parity-check (LDPC) codes Mathematical analysis Mathematical models Matrix message-passing algorithms Niobium Recovery Sampling, quantization Sensors Signal and communications theory Simulation sparse graphs Sparse matrices sparse sensing matrix success threshold Telecommunications and information theory verification-based recovery algorithms |
| Title | Density Evolution Analysis of Node-Based Verification-Based Algorithms in Compressed Sensing |
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