### Abstract

We consider the problem of recovering the superposition of R distinct complex exponential functions from compressed non-uniform time-domain samples. Total Variation (TV) minimization or atomic norm minimization was proposed in the literature to recover the R frequencies or the missing data. However, in order for TV minimization and atomic norm minimization to recover the missing data or the frequencies, the underlying R frequencies are required to be well-separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve the R complex exponentials and their frequencies from compressed nonuniform measurements, regardless of how close their frequencies are to each other. We propose a new concept of orthonormal atomic norm minimization (OANM), and demonstrate that the success of Hankel matrix recovery in separation-free super-resolution comes from the fact that the nuclear norm of a Hankel matrix is an orthonormal atomic norm. More specifically, we show that, in traditional atomic norm minimization, the underlying parameter values must be well separated to achieve successful signal recovery, if the atoms are changing continuously with respect to the continuously-valued parameter. In contrast, for the OANM, it is possible the OANM is successful even though the original atoms can be arbitrarily close.

Original language | English (US) |
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Title of host publication | 2018 IEEE International Symposium on Information Theory, ISIT 2018 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 76-80 |

Number of pages | 5 |

ISBN (Print) | 9781538647806 |

DOIs | |

State | Published - Jan 1 2018 |

Event | 2018 IEEE International Symposium on Information Theory, ISIT 2018 - Vail, United States Duration: Jun 17 2018 → Jun 22 2018 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2018-June |

ISSN (Print) | 2157-8095 |

### Other

Other | 2018 IEEE International Symposium on Information Theory, ISIT 2018 |
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Country | United States |

City | Vail |

Period | 6/17/18 → 6/22/18 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Information Systems
- Modeling and Simulation
- Applied Mathematics

### Cite this

*2018 IEEE International Symposium on Information Theory, ISIT 2018*(pp. 76-80). [8437560] (IEEE International Symposium on Information Theory - Proceedings; Vol. 2018-June). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT.2018.8437560

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*2018 IEEE International Symposium on Information Theory, ISIT 2018.*, 8437560, IEEE International Symposium on Information Theory - Proceedings, vol. 2018-June, Institute of Electrical and Electronics Engineers Inc., pp. 76-80, 2018 IEEE International Symposium on Information Theory, ISIT 2018, Vail, United States, 6/17/18. https://doi.org/10.1109/ISIT.2018.8437560

**Separation-Free Super-Resolution from Compressed Measurements is Possible : An Orthonormal Atomic Norm Minimization Approach.** / Xu, Weiyu; Yi, Jirong; Dasgupta, Soura; Cai, Jian Feng; Jacob, Mathews; Cho, Myung (Michael).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Separation-Free Super-Resolution from Compressed Measurements is Possible

T2 - An Orthonormal Atomic Norm Minimization Approach

AU - Xu, Weiyu

AU - Yi, Jirong

AU - Dasgupta, Soura

AU - Cai, Jian Feng

AU - Jacob, Mathews

AU - Cho, Myung (Michael)

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We consider the problem of recovering the superposition of R distinct complex exponential functions from compressed non-uniform time-domain samples. Total Variation (TV) minimization or atomic norm minimization was proposed in the literature to recover the R frequencies or the missing data. However, in order for TV minimization and atomic norm minimization to recover the missing data or the frequencies, the underlying R frequencies are required to be well-separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve the R complex exponentials and their frequencies from compressed nonuniform measurements, regardless of how close their frequencies are to each other. We propose a new concept of orthonormal atomic norm minimization (OANM), and demonstrate that the success of Hankel matrix recovery in separation-free super-resolution comes from the fact that the nuclear norm of a Hankel matrix is an orthonormal atomic norm. More specifically, we show that, in traditional atomic norm minimization, the underlying parameter values must be well separated to achieve successful signal recovery, if the atoms are changing continuously with respect to the continuously-valued parameter. In contrast, for the OANM, it is possible the OANM is successful even though the original atoms can be arbitrarily close.

AB - We consider the problem of recovering the superposition of R distinct complex exponential functions from compressed non-uniform time-domain samples. Total Variation (TV) minimization or atomic norm minimization was proposed in the literature to recover the R frequencies or the missing data. However, in order for TV minimization and atomic norm minimization to recover the missing data or the frequencies, the underlying R frequencies are required to be well-separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve the R complex exponentials and their frequencies from compressed nonuniform measurements, regardless of how close their frequencies are to each other. We propose a new concept of orthonormal atomic norm minimization (OANM), and demonstrate that the success of Hankel matrix recovery in separation-free super-resolution comes from the fact that the nuclear norm of a Hankel matrix is an orthonormal atomic norm. More specifically, we show that, in traditional atomic norm minimization, the underlying parameter values must be well separated to achieve successful signal recovery, if the atoms are changing continuously with respect to the continuously-valued parameter. In contrast, for the OANM, it is possible the OANM is successful even though the original atoms can be arbitrarily close.

UR - http://www.scopus.com/inward/record.url?scp=85052451601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052451601&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2018.8437560

DO - 10.1109/ISIT.2018.8437560

M3 - Conference contribution

AN - SCOPUS:85052451601

SN - 9781538647806

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 76

EP - 80

BT - 2018 IEEE International Symposium on Information Theory, ISIT 2018

PB - Institute of Electrical and Electronics Engineers Inc.

ER -