Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require less a priori knowledge, but their theoretical analysis is limited. In this paper, we propose and study a statistical machine learning approach, based on empirical risk minimization. Our main contribution is a theoretical analysis, showing that, provided with enough data, this approach can reach sharp rates while being essentially adaptive to the noise and smoothness of the problem. Numerical simulations corroborate and illustrate the theoretical findings. Our results are a step towards grounding theoretically data-driven approaches to inverse problems.

翻译：逆问题中最优正则化参数的选择是一个经典且具有挑战性的问题。近年来，数据驱动方法在解决这一挑战中日益流行。这类方法因所需先验知识较少而颇具吸引力，但其理论分析尚不充分。本文提出并研究了一种基于经验风险最小化的统计机器学习方法。我们的主要贡献在于理论分析，证明在数据量充足的条件下，该方法能够达到最优收敛速率，同时本质上对问题的噪声和光滑度具有自适应性。数值模拟验证并阐释了理论发现。本研究为逆问题中数据驱动方法的理论奠基迈出了重要一步。