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：在逆问题中选择最佳正则化参数是一个经典且富有挑战性的问题。近年来，数据驱动方法已成为应对这一挑战的流行手段。这类方法因其对先验知识需求较少而具有吸引力，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，表明在拥有足够数据的情况下，该方法能够达到锐利收敛速率，同时本质上对问题的噪声和平滑性具有自适应性。数值模拟验证并阐释了理论结果。本研究为数据驱动方法在逆问题中的理论奠基迈出了关键一步。