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Simple Error Bounds for Regularized Noisy Linear Inverse Problems

Thrampoulidis, Christos and Oymak, Samet and Hassibi, Babak (2014) Simple Error Bounds for Regularized Noisy Linear Inverse Problems. In: 2014 IEEE International Symposium on Information Theory (ISIT),. IEEE , Piscataway, NJ, pp. 3007-3011. ISBN 978-1-4799-5186-4. https://resolver.caltech.edu/CaltechAUTHORS:20150121-071420813

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Abstract

Consider estimating a structured signal x_0 from linear, underdetermined and noisy measurements y = Ax_0+z, via solving a variant of the lasso algorithm: x̂ = arg min_x{∥y-Ax∥+2 + λf(x)}. Here, f is a convex function aiming to promote the structure of x_0, say ℓ_1-norm to promote sparsity or nuclear norm to promote low-rankness. We assume that the entries of A are independent and normally distributed and make no assumptions on the noise vector z, other than it being independent of A. Under this generic setup, we derive a general, non-asymptotic and rather tight upper bound on the ℓ_2-norm of the estimation error ∥x̂ - x0∥2. Our bound is geometric in nature and obeys a simple formula; the roles of λ, f and x_0 are all captured by a single summary parameter δ(λ∂f(x_0)), termed the Gaussian squared distance to the scaled subdifferential. We connect our result to the literature and verify its validity through simulations.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1109/ISIT.2014.6875386DOIArticle
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6875386PublisherArticle
http://arxiv.org/abs/1401.6578arXivPreprint
ORCID:
AuthorORCID
Thrampoulidis, Christos0000-0001-9053-9365
Additional Information:© 2014 IEEE. This work was supported in part by the National Science Foundation under grants CCF-0729203, CNS-0932428 and CIF-1018927, by the Office of Naval Research under the MURI grant N00014-08-1-0747, and by a grant from Qualcomm Inc.
Funders:
Funding AgencyGrant Number
NSFCCF-0729203
NSFCNS-0932428
NSFCIF-1018927
Office of Naval Research (ONR)N00014-08-1-0747
Qualcomm Inc.UNSPECIFIED
Record Number:CaltechAUTHORS:20150121-071420813
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20150121-071420813
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:53909
Collection:CaltechAUTHORS
Deposited By: Shirley Slattery
Deposited On:22 Jan 2015 21:27
Last Modified:09 Mar 2020 13:19

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