Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for applications like low-dose CT or various sparse data problems. However, there are few theoretical results for deep learning in inverse problems. In this paper, we establish a complete convergence analysis for the proposed NETT (Network Tikhonov) approach to inverse problems. NETT considers data consistent solutions having small value of a regularizer defined by a trained neural network. We derive well-posedness results and quantitative error estimates, and propose a possible strategy for training the regularizer. Our theoretical results and framework are different from any previous work using neural networks for solving inverse problems. A possible data driven regularizer is proposed. Numerical results are presented for a tomographic sparse data problem, which demonstrate good performance of NETT even for unknowns of different type from the training data. To derive the convergence and convergence rates results we introduce a new framework based on the absolute Bregman distance generalizing the standard Bregman distance from the convex to the non-convex case.

中文翻译：

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NETT：用深度神经网络解决逆问题

从间接测量中恢复函数或高维参数向量是各个科学领域的核心任务。解决此类逆问题的几种方法已得到很好的发展和理解。最近，出现了使用深度学习和神经网络解决逆问题的新算法。虽然仍处于起步阶段，但这些技术在低剂量 CT 或各种稀疏数据问题等应用中表现出惊人的性能。然而，在逆问题中深度学习的理论结果很少。在本文中，我们为所提出的 NETT（Network Tikhonov）方法建立了一个完整的收敛分析来解决逆问题。NETT 认为数据一致的解决方案具有由训练有素的神经网络定义的正则化器的小值。我们推导出适定性结果和定量误差估计，并提出训练正则化器的可能策略。我们的理论结果和框架不同于之前使用神经网络解决逆问题的任何工作。提出了一种可能的数据驱动正则化器。给出了断层扫描稀疏数据问题的数值结果，即使对于来自训练数据的不同类型的未知数，也证明了 NETT 的良好性能。为了导出收敛和收敛速度结果，我们引入了一个基于绝对 Bregman 距离的新框架，概括了从凸面到非凸面情况的标准 Bregman 距离。给出了断层扫描稀疏数据问题的数值结果，即使对于来自训练数据的不同类型的未知数，也证明了 NETT 的良好性能。为了导出收敛和收敛速度结果，我们引入了一个基于绝对 Bregman 距离的新框架，概括了从凸面到非凸面情况的标准 Bregman 距离。给出了断层扫描稀疏数据问题的数值结果，即使对于来自训练数据的不同类型的未知数，也证明了 NETT 的良好性能。为了导出收敛和收敛速度结果，我们引入了一个基于绝对 Bregman 距离的新框架，概括了从凸面到非凸面情况的标准 Bregman 距离。