Deep neural networks have seen tremendous success over the last years. Since the training is performed on digital hardware, in this paper, we analyze what actually can be computed on current hardware platforms modeled as Turing machines, which would lead to inherent restrictions of deep learning. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. We prove that finite-dimensional inverse problems are not Banach-Mazur computable for small relaxation parameters. Even more, our results introduce a lower bound on the accuracy that can be obtained algorithmically.

翻译：近年来，深度神经网络取得了巨大成功。由于训练过程在数字硬件上进行，本文分析在建模为图灵机的当前硬件平台上实际能计算的内容，这将导致深度学习的固有限制。为此，我们聚焦于逆问题这一类问题，其特别涵盖任何从测量数据重建数据的任务。我们证明，对于小的松弛参数，有限维逆问题不是巴拿赫-马祖尔可计算的。更有甚者，我们的结果引入了算法可达到精度的下界。