Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors

Wei Zhang; Zhenni Wang

Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors

Wei Zhang
, Zhenni Wang

Department of Electrical Engineering

Research output: Chapters, Conference Papers, Creative and Literary Works › RGC 32 - Refereed conference paper (with host publication) › peer-review

1 Citation (Scopus)

Abstract

The restricted isometry property (RIP) is essential for the linear map to guarantee the successful recovery of low-rank matrices. The existing works show that the linear map generated by the measurement matrices with independent and identically distributed (i.i.d.) entries satisfies RIP with high probability. However, when dealing with non-i.i.d. measurement matrices, such as the rank-one measurements, the RIP compliance may not be guaranteed. In this paper, we show that the RIP can still be achieved with high probability, when the rank-one measurement matrix is constructed by the random unit-modulus vectors. Compared to the existing works, we first address the challenge of establishing RIP for the linear map in non-i.i.d. scenarios. As validated in the experiments, this linear map is memory-efficient, and not only satisfies the RIP but also exhibits similar recovery performance of the low-rank matrices to that of conventional i.i.d. measurement matrices. © 2024 by the author(s).

Original language	English
Title of host publication	Proceedings of the 27th International Conference on Artificial Intelligence and Statistics (AISTATS) 2024
Editors	Sanjoy Dasgupta, Stephan Mandt, Yingzhen Li
Publisher	ML Research Press
Pages	1900-1908
Publication status	Published - May 2024
Event	27th International Conference on Artificial Intelligence and Statistics (AISTATS 2024) - Palau de Congressos, Valencia, Spain Duration: 2 May 2024 → 4 May 2024 https://proceedings.mlr.press/v238/

Publication series

Name	Proceedings of Machine Learning Research
Volume	238
ISSN (Electronic)	2640-3498

Conference

Conference	27th International Conference on Artificial Intelligence and Statistics (AISTATS 2024)
Place	Spain
City	Valencia
Period	2/05/24 → 4/05/24
Internet address	https://proceedings.mlr.press/v238/

Bibliographical note

Full text of this publication does not contain sufficient affiliation information. With consent from the author(s) concerned, the Research Unit(s) information for this record is based on the existing academic department affiliation of the author(s).

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Zhang, W., & Wang, Z. (2024). Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors. In S. Dasgupta, S. Mandt, & Y. Li (Eds.), Proceedings of the 27th International Conference on Artificial Intelligence and Statistics (AISTATS) 2024 (pp. 1900-1908). (Proceedings of Machine Learning Research; Vol. 238). ML Research Press. https://proceedings.mlr.press/v238/zhang24d.html

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title = "Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors",

abstract = "The restricted isometry property (RIP) is essential for the linear map to guarantee the successful recovery of low-rank matrices. The existing works show that the linear map generated by the measurement matrices with independent and identically distributed (i.i.d.) entries satisfies RIP with high probability. However, when dealing with non-i.i.d. measurement matrices, such as the rank-one measurements, the RIP compliance may not be guaranteed. In this paper, we show that the RIP can still be achieved with high probability, when the rank-one measurement matrix is constructed by the random unit-modulus vectors. Compared to the existing works, we first address the challenge of establishing RIP for the linear map in non-i.i.d. scenarios. As validated in the experiments, this linear map is memory-efficient, and not only satisfies the RIP but also exhibits similar recovery performance of the low-rank matrices to that of conventional i.i.d. measurement matrices. {\textcopyright} 2024 by the author(s).",

author = "Wei Zhang and Zhenni Wang",

note = "Full text of this publication does not contain sufficient affiliation information. With consent from the author(s) concerned, the Research Unit(s) information for this record is based on the existing academic department affiliation of the author(s).; 27th International Conference on Artificial Intelligence and Statistics (AISTATS 2024) ; Conference date: 02-05-2024 Through 04-05-2024",

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month = may,

language = "English",

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Zhang, W & Wang, Z 2024, Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors. in S Dasgupta, S Mandt & Y Li (eds), Proceedings of the 27th International Conference on Artificial Intelligence and Statistics (AISTATS) 2024. Proceedings of Machine Learning Research, vol. 238, ML Research Press, pp. 1900-1908, 27th International Conference on Artificial Intelligence and Statistics (AISTATS 2024), Valencia, Spain, 2/05/24. <https://proceedings.mlr.press/v238/zhang24d.html>

Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors. / Zhang, Wei; Wang, Zhenni.
Proceedings of the 27th International Conference on Artificial Intelligence and Statistics (AISTATS) 2024. ed. / Sanjoy Dasgupta; Stephan Mandt; Yingzhen Li. ML Research Press, 2024. p. 1900-1908 (Proceedings of Machine Learning Research; Vol. 238).

Research output: Chapters, Conference Papers, Creative and Literary Works › RGC 32 - Refereed conference paper (with host publication) › peer-review

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AU - Zhang, Wei

AU - Wang, Zhenni

N1 - Full text of this publication does not contain sufficient affiliation information. With consent from the author(s) concerned, the Research Unit(s) information for this record is based on the existing academic department affiliation of the author(s).

PY - 2024/5

Y1 - 2024/5

N2 - The restricted isometry property (RIP) is essential for the linear map to guarantee the successful recovery of low-rank matrices. The existing works show that the linear map generated by the measurement matrices with independent and identically distributed (i.i.d.) entries satisfies RIP with high probability. However, when dealing with non-i.i.d. measurement matrices, such as the rank-one measurements, the RIP compliance may not be guaranteed. In this paper, we show that the RIP can still be achieved with high probability, when the rank-one measurement matrix is constructed by the random unit-modulus vectors. Compared to the existing works, we first address the challenge of establishing RIP for the linear map in non-i.i.d. scenarios. As validated in the experiments, this linear map is memory-efficient, and not only satisfies the RIP but also exhibits similar recovery performance of the low-rank matrices to that of conventional i.i.d. measurement matrices. © 2024 by the author(s).

AB - The restricted isometry property (RIP) is essential for the linear map to guarantee the successful recovery of low-rank matrices. The existing works show that the linear map generated by the measurement matrices with independent and identically distributed (i.i.d.) entries satisfies RIP with high probability. However, when dealing with non-i.i.d. measurement matrices, such as the rank-one measurements, the RIP compliance may not be guaranteed. In this paper, we show that the RIP can still be achieved with high probability, when the rank-one measurement matrix is constructed by the random unit-modulus vectors. Compared to the existing works, we first address the challenge of establishing RIP for the linear map in non-i.i.d. scenarios. As validated in the experiments, this linear map is memory-efficient, and not only satisfies the RIP but also exhibits similar recovery performance of the low-rank matrices to that of conventional i.i.d. measurement matrices. © 2024 by the author(s).

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T2 - 27th International Conference on Artificial Intelligence and Statistics (AISTATS 2024)

Y2 - 2 May 2024 through 4 May 2024

ER -

Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors

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