论文标题
线性统计反向学习问题中的最小二乘近似值
Least squares approximations in linear statistical inverse learning problems
论文作者
论文摘要
统计逆学习旨在从其他功能$ g $的随机分散和可能的嘈杂点评估中恢复未知功能$ f $，该功能通过一个不适合的数学模型连接到$ f $。在本文中，我们将统计逆学习理论与应用有限维度预测的经典正则化策略融合在一起。我们的主要发现是，将随机点评估的数量与投影维度的选择耦合，可以得出最大似然（ML）估计量的重建误差的概率收敛率。预期的收敛速率以ML估计量为基于规范的截止操作而得出。此外，我们证明所获得的速率是最小的。
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.
