Effective Condition Number Bounds for Convex Regularization

Amexlunxen, Dennis and Lotz, Martin and Walvin, Jake (2017) Effective Condition Number Bounds for Convex Regularization. [MIMS Preprint]

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Abstract

We derive bounds relating the statistical dimension of linear images of convex cones to Renegar's condition number. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection to a lower dimensional space, and can still be effective if the linear maps 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, interpreted as monotonicity property of moment functionals, and the kinematic formula from integral geometry. The main results are derived in the generalized setting of the biconic homogeneous feasibility problem.

Item Type: MIMS Preprint
Uncontrolled Keywords: Convex regularization; compressed sensing; random matrix theory; Gaussian width; condition numbers
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
MSC 2010, the AMS's Mathematics Subject Classification > 90 Operations research, mathematical programming
Depositing User: Dr. Martin Lotz
Date Deposited: 11 Sep 2017
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2572

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