# Asymptotic of Sparse Support Recovery for Positive Measures

Abstract : We study sparse spikes deconvolution over the space of Radon measures when the input measure is a finite sum of positive Dirac masses using the BLASSO convex program. We focus on the recovery properties of the support and the amplitudes of the initial measure in the presence of noise when the minimum separation distance t of the input measure (the minimum distance between two spikes) tends to zero. We show that when ||ω|| 2 /λ, ||ω|| 2 / t 2 N -1 and λ/ t 2 N -1 are small enough (where λ is the regularization parameter, ω the noise and N the number of spikes), which corresponds roughly to a sufficient signal-to-noise ratio and a noise level and a regularization parameter small enough with respect to the minimum separation distance, there exists a unique solution to the BLASSO program with exactly the same number of spikes as the original measure. We provide an upper bound on the error with respect to the initial measure. As a by-product, we show that the amplitudes and positions of the spikes of the solution both converge towards those of the input measure when λ and ω drop to zero faster than $t^{2N -1}$
Document type :
Conference papers

https://hal.archives-ouvertes.fr/hal-01271269
Contributor : Vincent Duval <>
Submitted on : Monday, February 8, 2016 - 10:53:18 PM
Last modification on : Wednesday, February 19, 2020 - 8:59:43 AM

### Citation

Quentin Denoyelle, Vincent Duval, Gabriel Peyré. Asymptotic of Sparse Support Recovery for Positive Measures. 5th International Workshop on New Computational Methods for Inverse Problems (NCMIP2015), 2015, Cachan, France. ⟨10.1088/1742-6596/657/1/012013⟩. ⟨hal-01271269⟩

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