参数反问题的神经网络正则化：拓扑问题
Regularization from examples via neural networks for parametric inverse problems: topology matters
摘要：
在这项工作中，我们处理参数逆问题，其中包括在恢复一个有限数量的参数描述一个未知对象的结构，从间接测量。用于通过使用前向模型的输入-输出对的数据集来近似正则化逆算子的现有技术方法依赖于深度学习技术。在这些方法中，训练神经网络以直接从数据预测所寻求的参数的值。在本文中，我们表明，这些方法提供次优的结果时，参数空间的拓扑结构是严格粗糙的比欧几里德。为了克服这个问题，我们提出了一个两步的策略，通过神经网络，它的工作原理在一般的拓扑条件下的正则化逆算子。首先将参数嵌入到低维欧几里得空间的一个子空间中，然后通过前向算子用神经网络逼近该子空间与参数空间的像之间的同胚.然后通过将嵌入的逆应用于网络预测来检索参数。结果显示的太阳耀斑的X射线成像与光谱仪/望远镜成像X射线的数据的问题。在这种情况下，参数空间是收缩为点的莫比乌斯带族。我们的模拟研究表明，使用神经网络预测的参数直接从数据产生系统误差由于非欧几里德拓扑的参数空间。该策略克服了拓扑问题，保证了重建的稳定性和准确性。
英文摘要：
In this work we deal with parametric inverse problems, which consist in recovering a finite number of parameters describing the structure of an unknown object, from indirect measurements. State-of-the-art methods for approximating a regularizing inverse operator by using a dataset of input-output pairs of the forward model rely on deep learning techniques. In these approaches, a neural network is trained to predict the value of the sought parameters directly from the data. In this paper, we show that these methods provide suboptimal results when the topology of the parameter space is strictly coarser than the Euclidean one. To overcome this issue, we propose a two-step strategy for approximating a regularizing inverse operator by means of a neural network, which works under general topological conditions. First, we embed the parameters into a subspace of a low-dimensional Euclidean space; second, we use a neural network to approximate a homeomorphism between the subspace and the image of the parameter space through the forward operator. The parameters are then retrieved by applying the inverse of the embedding to the network predictions. The results are shown for the problem of X-ray imaging of solar flares with data from the Spectrometer/Telescope for Imaging X-rays. In this case, the parameter space is a family of Moebius strips that collapse into a point. Our simulation studies show that the use of a neural network for predicting the parameters directly from the data yields systematic errors due to the non-Euclidean topology of the parameter space. The proposed strategy overcomes the topological issues and furnishes stable and accurate reconstructions.
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