Our focus is on robust recovery algorithms in statistical linear inverse problem. We consider two recovery routines - the much studied linear estimate originating from Kuks and Olman [42] and polyhedral estimate introduced in [37]. It was shown in [38] that risk of these estimates can be tightly upper-bounded for a wide range of a priori information about the model through solving a convex optimization problem, leading to a computationally efficient implementation of nearly optimal estimates of these types. The subject of the present paper is design and analysis of linear and polyhedral estimates which are robust with respect to the uncertainty in the observation matrix. We evaluate performance of robust estimates under stochastic and deterministic matrix uncertainty and show how the estimation risk can be bounded by the optimal value of efficiently solvable convex optimization problem; "presumably good" estimates of both types are then obtained through optimization of the risk bounds with respect to estimate parameters.

翻译：本文聚焦于统计线性逆问题中的稳健恢复算法。我们考虑两种恢复方法：一种是源自Kuks和Olman [42]的线性估计（该估计已被广泛研究），另一种是[37]中引入的多面体估计。已有研究[38]表明，通过求解凸优化问题，这些估计的风险可以在广泛的模型先验信息范围内得到严格的上界，从而使得这两类近最优估计的计算效率得以实现。本文的主题是设计和分析对观测矩阵不确定性具有稳健性的线性估计与多面体估计。我们评估了在随机与确定性矩阵不确定性下稳健估计的性能，并展示了如何利用可高效求解的凸优化问题的最优值来界定估计风险；随后，通过优化与估计参数相关的风险界，即可得到这两类"推测最优"的估计。