1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices

Xiyuan Wang; Kun Wang; Zhongshan Zhang; Keping Long

doi:10.1109/ICASSP.2016.7472551

1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices

Xiyuan Wang, Kun Wang, Zhongshan Zhang^*, Keping Long

^*Corresponding author for this work

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

Abstract

In this paper, we investigate the problem of recovering positive semi-definite (PSD) matrix from 1-bit sensing. The measurement matrix is rank-1 and constructed by the outer product of a pair of vectors, whose entries are independent and identically distributed (i.i.d.) Gaussian variables. The recovery problem is solved in closed form through a convex programming. Our analysis reveals that the solution is biased in general. However, in case of error-free measurement, we find that for rank-r PSD matrix with bounded condition number, the bias decreases with an order of O(1/r). Therefore, an approximate recovery is still possible. Numerical experiments are conducted to verify our analysis.

Original language	English
Title of host publication	2016 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016 - Proceedings
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	4613-4617
Number of pages	5
ISBN (Electronic)	9781479999880
DOIs	https://doi.org/10.1109/ICASSP.2016.7472551
Publication status	Published - 18 May 2016
Externally published	Yes
Event	41st IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016 - Shanghai, China Duration: 20 Mar 2016 → 25 Mar 2016

Publication series

Name	ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings
Volume	2016-May
ISSN (Print)	1520-6149

Conference

Conference	41st IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016
Country/Territory	China
City	Shanghai
Period	20/03/16 → 25/03/16

Keywords

1-bit compressed sensing
positive semi-definite matrix recovery
rank-1 measurement matrix
signal quantization

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10.1109/ICASSP.2016.7472551

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Wang, X., Wang, K., Zhang, Z., & Long, K. (2016). 1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices. In 2016 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016 - Proceedings (pp. 4613-4617). Article 7472551 (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings; Vol. 2016-May). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ICASSP.2016.7472551

Wang, Xiyuan ; Wang, Kun ; Zhang, Zhongshan et al. / 1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices. 2016 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016 - Proceedings. Institute of Electrical and Electronics Engineers Inc., 2016. pp. 4613-4617 (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings).

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title = "1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices",

abstract = "In this paper, we investigate the problem of recovering positive semi-definite (PSD) matrix from 1-bit sensing. The measurement matrix is rank-1 and constructed by the outer product of a pair of vectors, whose entries are independent and identically distributed (i.i.d.) Gaussian variables. The recovery problem is solved in closed form through a convex programming. Our analysis reveals that the solution is biased in general. However, in case of error-free measurement, we find that for rank-r PSD matrix with bounded condition number, the bias decreases with an order of O(1/r). Therefore, an approximate recovery is still possible. Numerical experiments are conducted to verify our analysis.",

keywords = "1-bit compressed sensing, positive semi-definite matrix recovery, rank-1 measurement matrix, signal quantization",

author = "Xiyuan Wang and Kun Wang and Zhongshan Zhang and Keping Long",

note = "Publisher Copyright: {\textcopyright} 2016 IEEE.; 41st IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016 ; Conference date: 20-03-2016 Through 25-03-2016",

year = "2016",

month = may,

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doi = "10.1109/ICASSP.2016.7472551",

language = "English",

series = "ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings",

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Wang, X, Wang, K, Zhang, Z & Long, K 2016, 1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices. in 2016 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016 - Proceedings., 7472551, ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, vol. 2016-May, Institute of Electrical and Electronics Engineers Inc., pp. 4613-4617, 41st IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016, Shanghai, China, 20/03/16. https://doi.org/10.1109/ICASSP.2016.7472551

1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices. / Wang, Xiyuan; Wang, Kun; Zhang, Zhongshan et al.
2016 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016 - Proceedings. Institute of Electrical and Electronics Engineers Inc., 2016. p. 4613-4617 7472551 (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings; Vol. 2016-May).

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

TY - GEN

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AU - Wang, Xiyuan

AU - Wang, Kun

AU - Zhang, Zhongshan

AU - Long, Keping

PY - 2016/5/18

Y1 - 2016/5/18

N2 - In this paper, we investigate the problem of recovering positive semi-definite (PSD) matrix from 1-bit sensing. The measurement matrix is rank-1 and constructed by the outer product of a pair of vectors, whose entries are independent and identically distributed (i.i.d.) Gaussian variables. The recovery problem is solved in closed form through a convex programming. Our analysis reveals that the solution is biased in general. However, in case of error-free measurement, we find that for rank-r PSD matrix with bounded condition number, the bias decreases with an order of O(1/r). Therefore, an approximate recovery is still possible. Numerical experiments are conducted to verify our analysis.

AB - In this paper, we investigate the problem of recovering positive semi-definite (PSD) matrix from 1-bit sensing. The measurement matrix is rank-1 and constructed by the outer product of a pair of vectors, whose entries are independent and identically distributed (i.i.d.) Gaussian variables. The recovery problem is solved in closed form through a convex programming. Our analysis reveals that the solution is biased in general. However, in case of error-free measurement, we find that for rank-r PSD matrix with bounded condition number, the bias decreases with an order of O(1/r). Therefore, an approximate recovery is still possible. Numerical experiments are conducted to verify our analysis.

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KW - positive semi-definite matrix recovery

KW - rank-1 measurement matrix

KW - signal quantization

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Wang X, Wang K, Zhang Z, Long K. 1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices. In 2016 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2016 - Proceedings. Institute of Electrical and Electronics Engineers Inc. 2016. p. 4613-4617. 7472551. (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings). doi: 10.1109/ICASSP.2016.7472551

1-Bit compressed sensing of positive semi-definite matrices via rank-1 measurement matrices

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