Support Recovery for Sparse Super-Resolution of Positive Measures
Support Recovery for Sparse Super-Resolution of Positive Measures
复制标题
支持积极措施稀疏超分辨率的恢复
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
G. Peyré
中科院分区：
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作者：
Quentin Denoyelle;V. Duval;G. Peyré
当输入测量是使用 BLASSO 凸程序的正狄拉克质量的有限和时，我们研究 $$\mathbb {R}$$R 或 $$\mathbb {T}$$T 上氡测量空间上的稀疏尖峰超分辨率。我们重点关注支撑的恢复特性以及存在噪声时初始测量的幅度，作为输入测量的最小间隔 t（两个尖峰之间的最小距离）的函数。我们证明，当 $${w}/\lambda $$w/λ、$${w}/t^{2N-1}$$w/t2N-1 和 $$\lambda /t^{2N-1}$$λ/t2N-1 足够小时（其中 $$\lambda $$λ 是正则化参数，w 是噪声，N 是尖峰数量），这大致相当于足够的信噪比和足够小的噪声水平最小分离，BLASSO 程序存在一个独特的解决方案，其尖峰数量与原始测量值完全相同。我们表明，当噪声和正则化参数比 $$t^{2N-1}$$t2N-1 更快地下降到零时，解的尖峰的幅度和位置都会向输入测量的幅度和位置收敛。
We study sparse spikes super-resolution over the space of Radon measures on $$\mathbb {R}$$R or $$\mathbb {T}$$T 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 as a function of the minimum separation t of the input measure (the minimum distance between two spikes). We show that when $${w}/\lambda $$w/λ, $${w}/t^{2N-1}$$w/t2N-1 and $$\lambda /t^{2N-1}$$λ/t2N-1 are small enough (where $$\lambda $$λ is the regularization parameter, w the noise and N the number of spikes), which corresponds roughly to a sufficient signal-to-noise ratio and a noise level small enough with respect to the minimum separation, there exists a unique solution to the BLASSO program with exactly the same number of spikes as the original measure. We show that the amplitudes and positions of the spikes of the solution both converge toward those of the input measure when the noise and the regularization parameter drops to zero faster than $$t^{2N-1}$$t2N-1.