Big in Japan: Regularizing Networks for Solving Inverse Problems

Abstract

Deep learning and (deep) neural networks are emerging tools to address inverse problems and image reconstruction tasks. Despite outstanding performance, the mathematical analysis for solving inverse problems by neural networks is mostly missing. In this paper, we introduce and rigorously analyze families of deep regularizing neural networks (RegNets) of the form B + N @A B , where B is a classical regularization and the network N @A B is trained to recover the missing part Id X B not found by the classical regularization. We show that these regularizing networks yield a convergent regularization method for solving inverse problems. Additionally, we derive convergence rates (quantitative error estimates) assuming a sufficient decay of the associated distance function. We demonstrate that our results recover existing convergence and convergence rates results for filter-based regularization methods as well as the recently introduced null space network as special cases. Numerical results are presented for a tomographic sparse data problem, which clearly demonstrate that the proposed RegNets improve the classical regularization as well as the null space network.In this paper, we go one step further and generalize the concept of deep null space learning by allowing the network to also act in the orthogonal complement of the null space of A in a controlled manner. This is in particular useful if the operator contains several small singular values that are not strictly equal to zero. Similar to the components in the kernel, these parts are difficult to be reconstructed by a classical linear regularization method and quantitative error estimates require strong smoothness assumptions on the objects to be recovered. Learning almost invisible components can significantly improve reconstruction results for less smooth objects.

Deep Learning of truncated singular values for limited view photoacoustic tomography

“…In this section we describe the proposed deep learning method for solving the discrete linear system (3). As it is shown in [11] the combination of a classical regularization method with deep CNNs yields a regularization method together with quantitative error estimates. Here we use the truncated SVD decomposition as classical regularization method and combine it with a deep network that recovers the truncated coefficients.…”

“…This includes CNNs in iterative schemes [5,6,1], or using a single CNN to improve an initial reconstruction [4,9,10]. Very recently, we have proven that sequences of suitable neural networks in combination with classical regularizations lead to convergent regularization methods [11]. In the present paper, we apply the concept of regularizing networks to the limited view problem of photoacoustic tomography (PAT).…”

An Efficient Quasi-Newton Method for Nonlinear Inverse Problems via Learned Singular Values

“…We then train a neural network to predict the associated singular values from the model outputs, making it possible to compute an approximate Jacobian without the need for numerical differentiation, resulting in the proposed neural network augmented Quasi-Newton method (NN-QN). In related studies, researchers have proposed learned SVD frameworks in applications to regularized inverse problems [25]- [27] We will first detail the mathematics and framework for the NN-QN approach in section II. We then present an application to the highly nonlinear inverse problem of electrical impedance tomography in section III.…”

Data-consistent neural networks for solving nonlinear inverse problems

“…In order to address the ill-posedness of linear inverse problems, regularizing networks of the form R a Φ G were introduced in [27,26]. Here G X Y 3 X defines any regularization and Φ X X 3 X are trained neural networks approximating data-consistent networks.…”