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Gridless Line Spectrum Estimation and Low-Rank Toeplitz Matrix Compression Using Structured Samplers: A Regularization-Free Approach

Qiao, Heng and Pal, Piya (2017) Gridless Line Spectrum Estimation and Low-Rank Toeplitz Matrix Compression Using Structured Samplers: A Regularization-Free Approach. IEEE Transactions on Signal Processing, 65 (9). pp. 2221-2236. ISSN 1053-587X. http://resolver.caltech.edu/CaltechAUTHORS:20170323-140917068

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Abstract

This paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank Toeplitz covariance matrix. Certain families of structured deterministic samplers are shown to efficiently compress a high-dimensional Toeplitz matrix of size N × N, producing a compressed sketch of size O(√r) × O(√r).The reconstruction problem can be cast as that of line spectrum estimation, whereby, in absence of noise, Toeplitz matrices of any size N can be exactly recovered from compressive sketches of size O(√r) × O(√r), no matter how large N is. In the presence of noise and finite data, the line spectrum estimation algorithm is combined with a novel denoising technique that only exploits a positive semidefinite (PSD) Toeplitz constraint to denoise the compressed sketch using a simple least-squares minimization framework. A major advantage of the algorithm is that it does not require any regularization parameter. It also enjoys lower computational complexity owing to its ability to predict the unobserved entries of the low-rank Toeplitz matrix. Explicit bounds on the reconstruction error are established and it is shown that the PSD constraint on the denoiser is sufficient to ensure stable reconstruction from a sketch of size O(√r) × O(√r). Extensive simulations demonstrate that the proposed algorithm provides better performance over random samplers and algorithms that use nuclear-norm-based regularizers


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1109/TSP.2017.2659644DOIArticle
http://ieeexplore.ieee.org/document/7833203/PublisherArticle
Additional Information:© 2017 IEEE. Manuscript received May 29, 2016; revised September 26, 2016, December 12, 2016, and January 11, 2017; accepted January 12, 2017. Date of publication January 25, 2017; date of current version February 21, 2017. The associate editor coordinating the review of this manuscript and approving it for publication was Dr.Wei Liu. This work was supported in part by the NSF CPS Synergy 1544798, in part by the University of Maryland, College Park, and in part by the University of California, San Diego.
Funders:
Funding AgencyGrant Number
NSFCNS-1544798
University of Maryland UNSPECIFIED
University of California, San DiegoUNSPECIFIED
Subject Keywords:Compressive covariance sampling, matrix sketching, off-grid compressed sensing, Toeplitz matrices, nested sampling, line spectrum estimation, Vandermonde decomposition lemma
Record Number:CaltechAUTHORS:20170323-140917068
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170323-140917068
Official Citation:H. Qiao and P. Pal, "Gridless Line Spectrum Estimation and Low-Rank Toeplitz Matrix Compression Using Structured Samplers: A Regularization-Free Approach," in IEEE Transactions on Signal Processing, vol. 65, no. 9, pp. 2221-2236, May1, 1 2017. doi: 10.1109/TSP.2017.2659644
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:75365
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:23 Mar 2017 21:26
Last Modified:23 Mar 2017 21:26

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