### Abstract

Exact reconstruction of a sparse signal for an under-determined linear system using the ℓ_{0}-measure is, in general, an NP-hard problem. The most popular approach is to relax the ℓ_{0}-optimization problem to an ℓ_{1}-approximation. However, the strength of this convex approximation relies upon rigid properties on the system, which are not verifiable in practice. Greedy algorithms have been proposed in the past to speed up the optimization of the ℓ_{1} problem, but their computational efficiency comes at the expense of a larger error. In an effort to control error and complexity, this paper goes beyond the ℓ_{1}-approximation by growing neighborhoods of the ℓ_{1}-solution that moves towards the optimal solution. The size of the neighborhood is tunable depending on the computational resources. The proposed algorithm, termed Approximate Kernel RecONstruction (AKRON), yields significantly smaller errors than current greedy methods with a controllable computational cost. By construction, the error of AKRON is smaller than or to equal the ℓ_{1}-solution. AKRON enjoys all the error bounds of ℓ_{1} under the restricted isometry property condition. We benchmarked AKRON on simulated data from several under-determined systems, and the results show that AKRON can significantly improve the reconstruction error with slightly more computational cost than solving the ℓ_{1} problem directly.

Original language | English (US) |
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Pages (from-to) | 265-270 |

Number of pages | 6 |

Journal | Signal Processing |

Volume | 144 |

DOIs | |

Publication status | Published - Mar 2018 |

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### Keywords

- Compressive sensing
- Linear systems
- Optimization

### ASJC Scopus subject areas

- Control and Systems Engineering
- Software
- Signal Processing
- Computer Vision and Pattern Recognition
- Electrical and Electronic Engineering

### Cite this

*Signal Processing*,

*144*, 265-270. https://doi.org/10.1016/j.sigpro.2017.10.020