  In recent years, new regularization methods based on (deep) neural networks
have shown very promising empirical performance for the numerical solution of
ill-posed problems, e.g., in medical imaging and imaging science. Due to the
nonlinearity of neural networks, these methods often lack satisfactory
theoretical justification. In this work, we rigorously discuss the convergence
of a successful unsupervised approach that utilizes untrained convolutional
neural networks to represent solutions to linear ill-posed problems. Untrained
neural networks are particularly appealing for many applications because they
do not require paired training data. The regularization property of the
approach relies solely on the architecture of the neural network instead. Due
to the vast over-parameterization of the employed neural network, suitable
early stopping is essential for the success of the method. We establish that
the classical discrepancy principle is an adequate method for early stopping of
two-layer untrained convolutional neural networks learned by gradient descent,
and furthermore, it yields an approximation with minimax optimal convergence
rates. Numerical results are also presented to illustrate the theoretical
findings.
