# Regularization from examples via neural networks for parametric inverse problems: topology matters

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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