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Sparse Covariance Estimation from Quadratic Measurements: A Precise Analysis

Abbasi, Ehsan and Salehi, Fariborz and Hassibi, Babak (2019) Sparse Covariance Estimation from Quadratic Measurements: A Precise Analysis. In: 2019 IEEE International Symposium on Information Theory (ISIT). IEEE , Piscataway, NJ, pp. 2074-2078. ISBN 9781538692912.

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We study the problem of estimating a high-dimensional sparse covariance matrix, Σ_0, from a finite number of quadratic measurements, i.e., measurements a^T_iΣ_0_ai which are quadratic forms in the measurement vectors a i resulting from the covariance matrix, Σ_0. Such a problem arises in applications where we can only make energy measurements of the underlying random variables. We study a simple LASSO-like convex recovery algorithm which involves a squared 2-norm (to match the covariance estimate to the measurements), plus a regularization term (that penalizes the ℓ_1−norm of the non-diagonal entries of Σ_0 to enforce sparsity). When the measurement vectors are i.i.d. Gaussian, we obtain the precise error performance of the algorithm (accurately determining the estimation error in any metric, e.g., 2-norm, operator norm, etc.) as a function of the number of measurements and the underlying distribution of Σ_0. In particular, in the noiseless case we determine the necessary and sufficient number of measurements required to perfectly recover Σ_0 as a function of its sparsity. Our results rely on a novel comparison lemma which relates a convex optimization problem with "quadratic Gaussian" measurements to one which has i.i.d. Gaussian measurements.

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Additional Information:© 2019 IEEE.
Record Number:CaltechAUTHORS:20191004-100332908
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Official Citation:E. Abbasi, F. Salehi and B. Hassibi, "Sparse Covariance Estimation from Quadratic Measurements: A Precise Analysis," 2019 IEEE International Symposium on Information Theory (ISIT), Paris, France, 2019, pp. 2074-2078. doi: 10.1109/ISIT.2019.8849405
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:99075
Deposited By: George Porter
Deposited On:04 Oct 2019 18:14
Last Modified:16 Nov 2021 17:43

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