Learning-based methods for inverse problems, adapting to the data's inherent structure, have become ubiquitous in the last decade. Besides empirical investigations of their often remarkable performance, an increasing number of works addresses the issue of theoretical guarantees. Recently, [3] exploited invertible residual networks (iResNets) to learn provably convergent regularizations given reasonable assumptions. They enforced these guarantees by approximating the linear forward operator with an iResNet. Supervised training on relevant samples introduces data dependency into the approach. An open question in this context is to which extent the data's inherent structure influences the training outcome, i.e., the learned reconstruction scheme. Here we address this delicate interplay of training design and data dependency from a Bayesian perspective and shed light on opportunities and limitations. We resolve these limitations by analyzing reconstruction-based training of the inverses of iResNets, where we show that this optimization strategy introduces a level of data-dependency that cannot be achieved by approximation training. We further provide and discuss a series of numerical experiments underpinning and extending the theoretical findings.

翻译：基于学习的逆问题方法能够适应数据的固有结构，在过去十年中已变得无处不在。除了对其显著性能的实证研究外，越来越多的成果开始关注理论保证问题。近期，[3]利用可逆残差网络（iResNets）在合理假设下学习具有可收敛性的正则化方法。他们通过用iResNet近似线性正向算子来强制执行这些保证，并在相关样本上的监督训练将数据依赖性引入该方法。在此背景下，一个开放性问题在于数据固有结构能在多大程度上影响训练结果（即学习到的重建方案）。本文从贝叶斯视角探讨训练设计与数据依赖性之间微妙的相互作用，阐明其机遇与局限性。通过分析基于重建的iResNet逆网络训练，我们解决了这些局限性：研究表明，这种优化策略引入的数据依赖程度是近似训练无法实现的。此外，我们提供并讨论了一系列数值实验，这些实验佐证并扩展了理论发现。