In recent years, new regularization methods based on (deep) neural networks have shown very promising empirical performance for the numerical solution of ill-posed problems, such as in medical imaging and imaging science. Due to the nonlinearity of neural networks, these methods often lack satisfactory theoretical justification. In this work, we rigorously discuss the convergence of a successful unsupervised approach that utilizes untrained convolutional neural networks to represent solutions to linear ill-posed problems. Untrained neural networks have particular appeal for many applications because they do not require paired training data. The regularization property of the approach relies solely on the architecture of the neural network instead. Due to the vast over-parameterization of the employed neural network, suitable early stopping is essential for the success of the method. We establish that the classical discrepancy principle is an adequate method for early stopping of two-layer untrained convolutional neural networks learned by gradient descent, and furthermore, it yields an approximation with minimax optimal convergence rates. Numerical results are also presented to illustrate the theoretical findings.

翻译：近年来，基于（深度）神经网络的新型正则化方法在求解医学成像、成像科学等不适定问题的数值解中展现出极具前景的经验性能。由于神经网络的非线性特性，这类方法往往缺乏令人满意的理论依据。本研究严格论证了一种成功无监督方法的收敛性——该方法利用未训练卷积神经网络表示线性不适定问题的解。未训练神经网络因无需成对训练数据，在众多应用中具有特殊吸引力。该方法的正则化特性完全依赖于神经网络架构本身。由于所用神经网络的过度参数化特性，合适的早停策略对方法成功至关重要。我们证明经典偏差原则是梯度下降训练的两层未训练卷积神经网络的有效早停方法，且能产生具有极小极大最优收敛速度的近似解。文中还通过数值结果验证了理论发现。