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：统计逆学习旨在从另一个函数$g$的随机散点且可能含噪的点值评估中恢复未知函数$f$，其中$g$通过一个不适定数学模型与$f$相关联。本文将统计逆学习理论与应用有限维投影的正则化经典策略相结合。我们的关键发现是，通过将随机点评估的数量与投影维度的选择相耦合，可以推导出最大似然（ML）估计器重构误差的概率收敛率。通过引入基于范数的截断操作对ML估计器进行补充，我们进一步推导了期望下的收敛率。此外，我们证明所获得的速率是极小化最优的。