This paper studies the identification of a linear combination of point sources from a finite number of measurements. Since the data are typically contaminated by Gaussian noise, a statistical framework for its recovery is considered. It relies on two main ingredients, first, a convex but non-smooth Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger-Kantorovich distance to the ground truth. Assuming standard non-degenerate source conditions as well as applying careful linearization arguments, a computable upper bound on the latter is derived. On the one hand, this allows to derive asymptotic convergence results for the mean-squared error of the estimator in the small small variance case. On the other, it paves the way for applying optimal sensor placement approaches to sparse inverse problems.

翻译：本文研究了从有限次测量中识别点源线性组合的问题。由于数据通常受高斯噪声污染，我们考虑采用统计框架进行恢复。该框架依赖于两个主要要素：首先，在Radon测度空间上采用凸但非光滑的Tikhonov点估计量；其次，基于该估计量与真实值之间的Hellinger-Kantorovich距离构建合适的均方误差。在标准非退化源条件假设下，通过精细的线性化分析，推导出该均方误差的可计算上界。一方面，该上界可用于推导小方差情形下估计量均方误差的渐近收敛结果；另一方面，它为将最优传感器布局方法应用于稀疏反问题铺平了道路。