Separation-Free Super-Resolution from Compressed Measurements is Possible: An Orthonormal Atomic Norm Minimization Approach

Weiyu Xu, Jirong Yi, Soura Dasgupta, Jian Feng Cai, Mathews Jacob, Myung (Michael) Cho

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publication2018 IEEE International Symposium on Information Theory, ISIT 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages76-80
Number of pages5
ISBN (Print)9781538647806
DOIs
StatePublished - Jan 1 2018
Event2018 IEEE International Symposium on Information Theory, ISIT 2018 - Vail, United States
Duration: Jun 17 2018Jun 22 2018

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2018-June
ISSN (Print)2157-8095

Other

Other2018 IEEE International Symposium on Information Theory, ISIT 2018
CountryUnited States
CityVail
Period6/17/186/22/18

Fingerprint

Super-resolution
Orthonormal
Norm
Hankel Matrix
Recovery
Total Variation
Missing Data
Atoms
Exponential functions
Complex Functions
Superposition
Time Domain
Resolve
Distinct
Demonstrate

All Science Journal Classification (ASJC) codes

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

Cite this

Xu, W., Yi, J., Dasgupta, S., Cai, J. F., Jacob, M., & Cho, M. M. (2018). Separation-Free Super-Resolution from Compressed Measurements is Possible: An Orthonormal Atomic Norm Minimization Approach. In 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
Xu, Weiyu ; Yi, Jirong ; Dasgupta, Soura ; Cai, Jian Feng ; Jacob, Mathews ; Cho, Myung (Michael). / Separation-Free Super-Resolution from Compressed Measurements is Possible : An Orthonormal Atomic Norm Minimization Approach. 2018 IEEE International Symposium on Information Theory, ISIT 2018. Institute of Electrical and Electronics Engineers Inc., 2018. pp. 76-80 (IEEE International Symposium on Information Theory - Proceedings).
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Xu, W, Yi, J, Dasgupta, S, Cai, JF, Jacob, M & Cho, MM 2018, Separation-Free Super-Resolution from Compressed Measurements is Possible: An Orthonormal Atomic Norm Minimization Approach. in 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).

2018 IEEE International Symposium on Information Theory, ISIT 2018. Institute of Electrical and Electronics Engineers Inc., 2018. p. 76-80 8437560 (IEEE International Symposium on Information Theory - Proceedings; Vol. 2018-June).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Xu W, Yi J, Dasgupta S, Cai JF, Jacob M, Cho MM. Separation-Free Super-Resolution from Compressed Measurements is Possible: An Orthonormal Atomic Norm Minimization Approach. In 2018 IEEE International Symposium on Information Theory, ISIT 2018. Institute of Electrical and Electronics Engineers Inc. 2018. p. 76-80. 8437560. (IEEE International Symposium on Information Theory - Proceedings). https://doi.org/10.1109/ISIT.2018.8437560