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：学习型反问题求解器在图像重建等应用中展现出卓越性能。这类数据驱动重建方法通常遵循两步框架：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数据以获得重建结果。我们遵循该流程但采用可逆残差网络（iResNets）参数化重建方案。研究表明，可逆性使得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特性的条件下，我们证明这类方案可成为收敛正则化方法。此外，研究揭示了其与线性反问题正则化理论的形式化联系，并针对特定架构类别提出了非线性谱正则化方法。在数值实验方面，我们验证了选定训练架构的局部逼近特性，并在MNIST数据集上开展系列实验，这些实验不仅验证了理论发现，更拓展了其适用范围。