Asymptotic of Sparse Support Recovery for Positive Measures

5th International Workshop on New Computational Methods for Inverse Problems

物理学;计算机科学

Denoyelle, Q.^1 ; Duval, V.^1 ; Peyré, G.^1

CEREMADE, Université Paris-Dauphine INRIA CEREMADE, CNRS, Université Paris-Dauphine, France^1

关键词: Convex programs;    Minimum distance;    Noise levels;    Radon measure;    Recovery properties;    Regularization parameters;    Separation distances;    Support recoveries;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/657/1/012013/pdf
DOI  :  10.1088/1742-6596/657/1/012013

学科分类：计算机科学（综合）

来源: IOP

【 摘 要 】

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/t2N-1and λ/t2N-1are 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 t2N-1.