未知算子反问题的深度学习
Deep learning for inverse problems with unknown operator
摘要：
我们考虑不适定的逆问题，其中正向算子$T$是未知的，相反，我们可以访问由函数$f_i$和它们的噪声图像$Tf_i$组成的训练数据。这是一个实际相关且具有挑战性的问题，当前的方法只能在对训练集的强假设下才能解决。在这里，我们提出了一种新的方法，需要对数据的假设最少，并证明重建率取决于训练点的数量和噪声水平。我们表明，在“许多”训练数据的制度，该方法是最小最大最优的。所提出的方法采用了一种卷积神经网络（U网）和经验风险最小化，以“适合”未知的运营商。简而言之，我们的方法基于两个想法：第一个是将U-网与小波等多尺度分解联系起来，从而将它们与现有的理论联系起来，第二个是使用U-网的层次结构和卷积神经网络的低参数来证明实际有用的熵界。与现有的非参数统计中神经网络的工作的一个显着区别是，我们使用它们来近似运算符，而不是函数，我们认为这在数学上更自然，技术上更方便。
英文摘要：
We consider ill-posed inverse problems where the forward operator $T$ is unknown, and instead we have access to training data consisting of functions $f_i$ and their noisy images $Tf_i$. This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of "many" training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to "fit" the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.
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