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Low noise regimes for ℓ regularization : continuous and discrete settings

Duval, Vincent; Peyré, Gabriel (2014), Low noise regimes for ℓ regularization : continuous and discrete settings, Proceedings in Applied Mathematics and Mechanics, 14, 1, p. 943–944. http://dx.doi.org/10.1002/pamm.201410452

Type
Article accepté pour publication ou publié
Date
2014
Journal name
Proceedings in Applied Mathematics and Mechanics
Volume
14
Number
1
Publisher
Wiley
Pages
943–944
Publication identifier
http://dx.doi.org/10.1002/pamm.201410452
Metadata
Show full item record
Author(s)
Duval, Vincent cc
Peyré, Gabriel
Abstract (EN)
We focus on support recovery for signal deconvolution with sparsity assumption. We adopt the continuous setting defined by several recent works and we try to reconstruct a sum of Dirac masses from its low frequencies (possibly perturbed by some noise), by using a total variation prior for Radon measures (i.e. the generalization to measures of the ℓ1 norm). We show that, under a non degenerate source condition, there exists a small noise regime in which the model recovers exactly the same number of spikes as the original signal, and the spikes converge to those of the original signal as the noise vanishes. This continuous setting, by allowing the spikes to “move”, provides robust support recovery for signals composed of well separated spikes. In a discrete setting, where the spikes are reconstructed on a grid, similar low noise regimes which guarantee the exact recovery of the support also exist (see [3]). Yet, this property only concerns a small class of signals. Considering the asymptotics of the discrete problems as the size of the grid tends to zero, we show that the support of the original signal cannot be stable on thin grids, and that the discrete models actually reconstruct pairs of spikes near each original spike. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
Subjects / Keywords
Low noise regimes

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