Compressed Sensing with nonlinear observations and related non-linear optimisation problems
Compressed Sensing with nonlinear observations and related non-linear optimisation problems
Non-convex constraints have recently proven a valuable tool in many optimisation problems. In particular sparsity constraints have had a significant impact on sampling theory, where they are used in Compressed Sensing and allow structured signals to be sampled far below the rate traditionally prescribed.
Nearly all of the theory developed for Compressed Sensing signal recovery assumes that samples are taken using linear measurements. In this paper we instead address the Compressed Sensing recovery problem in a setting where the observations are non-linear. We show that, under conditions similar to those required in the linear setting, the Iterative Hard Thresholding algorithm can be used to accurately recover sparse or structured signals from few non-linear observations.
Similar ideas can also be developed in a more general non-linear optimisation framework. In the second part of this paper we therefore present related result that show how this can be done under sparsity and union of subspaces constraints, whenever a generalisation of the Restricted Isometry Property traditionally imposed on the Compressed Sensing system holds.
compressed sensing, nonlinear optimisation, non-convex constraints, inverse problems
3466-3474
Blumensath, Thomas
470d9055-0373-457e-bf80-4389f8ec4ead
22 February 2013
Blumensath, Thomas
470d9055-0373-457e-bf80-4389f8ec4ead
Blumensath, Thomas
(2013)
Compressed Sensing with nonlinear observations and related non-linear optimisation problems.
IEEE Trans Information Theory, 59 (6), .
(doi:10.1109/TIT.2013.2245716).
Abstract
Non-convex constraints have recently proven a valuable tool in many optimisation problems. In particular sparsity constraints have had a significant impact on sampling theory, where they are used in Compressed Sensing and allow structured signals to be sampled far below the rate traditionally prescribed.
Nearly all of the theory developed for Compressed Sensing signal recovery assumes that samples are taken using linear measurements. In this paper we instead address the Compressed Sensing recovery problem in a setting where the observations are non-linear. We show that, under conditions similar to those required in the linear setting, the Iterative Hard Thresholding algorithm can be used to accurately recover sparse or structured signals from few non-linear observations.
Similar ideas can also be developed in a more general non-linear optimisation framework. In the second part of this paper we therefore present related result that show how this can be done under sparsity and union of subspaces constraints, whenever a generalisation of the Restricted Isometry Property traditionally imposed on the Compressed Sensing system holds.
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Published date: 22 February 2013
Keywords:
compressed sensing, nonlinear optimisation, non-convex constraints, inverse problems
Organisations:
Signal Processing & Control Grp
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Local EPrints ID: 164753
URI: http://eprints.soton.ac.uk/id/eprint/164753
PURE UUID: fcda1b11-b404-4e7a-a458-b3496da13960
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Date deposited: 05 Oct 2010 07:46
Last modified: 14 Mar 2024 02:55
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