Low-Rank Covariance Matrix Recovery From Rank-One Measurements: An Analytical Solution

Peilan Wang; Jun Fang; Binyao Ma; Bin Wang; Geert Leus

doi:10.1109/LSP.2025.3584252

Low-Rank Covariance Matrix Recovery From Rank-One Measurements: An Analytical Solution

Peilan Wang, Jun Fang^*, Binyao Ma, Bin Wang, Geert Leus

^*Corresponding author for this work

Signal Processing Systems

Research output: Contribution to journal › Article › Scientific › peer-review

Abstract

In this letter, we propose an analytical solution for recovering a low-rank positive semi-definite (PSD) matrix from its rank-one measurements. We show that by utilizing a set of structured measurement vectors, we can analytically determine the null space of this low-rank PSD matrix. Based on the result, the PSD matrix can be efficiently recovered. Our analysis shows that the proposed method only requires (N - K)(2K + 1)+ K ² measurements to guarantee exact recovery of the PSD matrix, where N and K respectively denote the dimension and the rank of the PSD matrix. Numerical results show that the proposed method achieves a considerable improvement over existing state-of-the-art methods in terms of both sample complexity and computational efficiency. Specifically, the proposed method helps improve the computational efficiency by an order of magnitude as compared with existing methods.

Original language	English
Pages (from-to)	2674-2678
Number of pages	5
Journal	IEEE Signal Processing Letters
Volume	32
DOIs	https://doi.org/10.1109/LSP.2025.3584252
Publication status	Published - 2025

Bibliographical note

Green Open Access added to TU Delft Institutional Repository as part of the Taverne amendment. More information about this copyright law amendment can be found at https://www.openaccess.nl. Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Keywords

covariance sketching
Low-rank covariance matrix recovery
rank-one measurements

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title = "Low-Rank Covariance Matrix Recovery From Rank-One Measurements: An Analytical Solution",

abstract = "In this letter, we propose an analytical solution for recovering a low-rank positive semi-definite (PSD) matrix from its rank-one measurements. We show that by utilizing a set of structured measurement vectors, we can analytically determine the null space of this low-rank PSD matrix. Based on the result, the PSD matrix can be efficiently recovered. Our analysis shows that the proposed method only requires (N - K)(2K + 1)+ K 2 measurements to guarantee exact recovery of the PSD matrix, where N and K respectively denote the dimension and the rank of the PSD matrix. Numerical results show that the proposed method achieves a considerable improvement over existing state-of-the-art methods in terms of both sample complexity and computational efficiency. Specifically, the proposed method helps improve the computational efficiency by an order of magnitude as compared with existing methods.",

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author = "Peilan Wang and Jun Fang and Binyao Ma and Bin Wang and Geert Leus",

note = "Green Open Access added to TU Delft Institutional Repository as part of the Taverne amendment. More information about this copyright law amendment can be found at https://www.openaccess.nl. Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.",

year = "2025",

doi = "10.1109/LSP.2025.3584252",

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T1 - Low-Rank Covariance Matrix Recovery From Rank-One Measurements

T2 - An Analytical Solution

AU - Wang, Peilan

AU - Fang, Jun

AU - Ma, Binyao

AU - Wang, Bin

AU - Leus, Geert

N1 - Green Open Access added to TU Delft Institutional Repository as part of the Taverne amendment. More information about this copyright law amendment can be found at https://www.openaccess.nl. Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

PY - 2025

Y1 - 2025

N2 - In this letter, we propose an analytical solution for recovering a low-rank positive semi-definite (PSD) matrix from its rank-one measurements. We show that by utilizing a set of structured measurement vectors, we can analytically determine the null space of this low-rank PSD matrix. Based on the result, the PSD matrix can be efficiently recovered. Our analysis shows that the proposed method only requires (N - K)(2K + 1)+ K 2 measurements to guarantee exact recovery of the PSD matrix, where N and K respectively denote the dimension and the rank of the PSD matrix. Numerical results show that the proposed method achieves a considerable improvement over existing state-of-the-art methods in terms of both sample complexity and computational efficiency. Specifically, the proposed method helps improve the computational efficiency by an order of magnitude as compared with existing methods.

AB - In this letter, we propose an analytical solution for recovering a low-rank positive semi-definite (PSD) matrix from its rank-one measurements. We show that by utilizing a set of structured measurement vectors, we can analytically determine the null space of this low-rank PSD matrix. Based on the result, the PSD matrix can be efficiently recovered. Our analysis shows that the proposed method only requires (N - K)(2K + 1)+ K 2 measurements to guarantee exact recovery of the PSD matrix, where N and K respectively denote the dimension and the rank of the PSD matrix. Numerical results show that the proposed method achieves a considerable improvement over existing state-of-the-art methods in terms of both sample complexity and computational efficiency. Specifically, the proposed method helps improve the computational efficiency by an order of magnitude as compared with existing methods.

KW - covariance sketching

KW - Low-rank covariance matrix recovery

KW - rank-one measurements

UR - https://www.scopus.com/pages/publications/105010074959

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DO - 10.1109/LSP.2025.3584252

M3 - Article

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SN - 1070-9908

VL - 32

SP - 2674

EP - 2678

JO - IEEE Signal Processing Letters

JF - IEEE Signal Processing Letters

ER -

Low-Rank Covariance Matrix Recovery From Rank-One Measurements: An Analytical Solution

Abstract

Bibliographical note

Keywords

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