Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization strategy of applying finite-dimensional projections. Our key finding is that coupling the number of random point evaluations with the choice of projection dimension, one can derive probabilistic convergence rates for the reconstruction error of the maximum likelihood (ML) estimator. Convergence rates in expectation are derived with a ML estimator complemented with a norm-based cut-off operation. Moreover, we prove that the obtained rates are minimax optimal.

翻译：统计反常学习旨在从随机分散和可能吵闹的对另一个函数(g)美元的评价中回收一个未知的功能(ff$美元),该功能通过一个错误的数学模型连接到f$美元。在本文中,我们把统计反向学习理论与应用有限维预测的传统正规化战略相结合。我们的主要发现是,将随机点评价的数量与预测层面的选择结合起来,人们可以得出最大可能性(ML)估计测量仪重建错误的概率趋同率。预期的趋同率与ML估计值相结合,并辅之以基于规范的截断操作。此外,我们证明所获得的比率是最理想的。