The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational 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. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

翻译：本文旨在量化稀疏逆问题中带随机噪声的测度重建误差，并以此驱动最优传感器配置研究。为实现该目标，误差量需显式依赖于传感器配置且对源项具有鲁棒性，同时相比通过大量样本直接评估误差，更需具备实际计算可行性。具体而言，我们研究从受高斯噪声污染的有限测量中识别由点源未知线性组合构成的测度问题。恢复的统计框架基于两个核心要素：其一，在拉东测度空间上构建凸但非光滑的变分吉洪诺夫点估计器；其二，基于估计量与真值之间的赫林格-坎托罗维奇距离建立合适的均方误差准则。为量化误差，我们引入非退化源条件并通过精细线性化推导出可计算的上界，最终获得传感器配置显式表达的渐近紧误差期望估计。该结果既可评估给定传感器配置下的预期重建误差，亦可指导稀疏逆问题中的传感器优化布局。