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Effective condition number bounds for convex regularization

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Amelunxen, Dennis, Lotz, Martin and Walvin, Jake (2020) Effective condition number bounds for convex regularization. IEEE Transactions on Information Theory, 66 (4). 2501 -2516. doi:10.1109/TIT.2020.2965720

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Official URL: https://doi.org/10.1109/TIT.2020.2965720

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Abstract

We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the ℓ 1 -analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality and the kinematic formula from integral geometry.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Convex sets, Integral geometry, Mathematical optimization
Journal or Publication Title: IEEE Transactions on Information Theory
Publisher: IEEE
ISSN: 0018-9448
Official Date: April 2020
Dates:
DateEvent
April 2020Published
10 January 2020Available
23 September 2019Accepted
Volume: 66
Number: 4
Page Range: 2501 -2516
DOI: 10.1109/TIT.2020.2965720
Status: Peer Reviewed
Publication Status: Published
Publisher Statement: © 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Access rights to Published version: Restricted or Subscription Access
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