Learned inverse problem solvers exhibit remarkable performance in applications like image reconstruction tasks. These data-driven reconstruction methods often follow a two-step scheme. First, one trains the often neural network-based reconstruction scheme via a dataset. Second, one applies the scheme to new measurements to obtain reconstructions. We follow these steps but parameterize the reconstruction scheme with invertible residual networks (iResNets). We demonstrate that the invertibility enables investigating the influence of the training and architecture choices on the resulting reconstruction scheme. For example, assuming local approximation properties of the network, we show that these schemes become convergent regularizations. In addition, the investigations reveal a formal link to the linear regularization theory of linear inverse problems and provide a nonlinear spectral regularization for particular architecture classes. On the numerical side, we investigate the local approximation property of selected trained architectures and present a series of experiments on the MNIST dataset that underpin and extend our theoretical findings.

翻译：通过学习得到的逆问题求解器在图像重建等应用领域展现出卓越性能。这类数据驱动重建方法通常遵循两步方案：首先通过数据集训练基于神经网络的重建方案，随后将该方案应用于新测量值以获得重建结果。本文遵循上述步骤，但采用可逆残差网络（iResNets）参数化重建方案。研究表明，可逆性使我们能够探究训练过程和架构选择对最终重建方案的影响。例如，在假设网络具有局部逼近性质的前提下，我们证明此类方案可成为收敛正则化方法。此外，研究揭示了与线性逆问题线性正则化理论的形式化关联，并为特定架构类别提供了非线性谱正则化方法。在数值实验方面，我们探究了选定训练架构的局部逼近性质，并在MNIST数据集上开展了一系列实验，这些实验验证并拓展了我们的理论发现。