  Inverse problems arise in a number of domains such as medical imaging, remote
sensing, and many more, relying on the use of advanced signal and image
processing approaches -- such as sparsity-driven techniques -- to determine
their solution. This paper instead studies the use of deep learning approaches
to approximate the solution of inverse problems. In particular, the paper
provides a new generalization bound, depending on key quantity associated with
a deep neural network -- its Jacobian matrix -- that also leads to a number of
computationally efficient regularization strategies applicable to inverse
problems. The paper also tests the proposed regularization strategies in a
number of inverse problems including image super-resolution ones. Our numerical
results conducted on various datasets show that both fully connected and
convolutional neural networks regularized using the regularization or proxy
regularization strategies originating from our theory exhibit much better
performance than deep networks regularized with standard approaches such as
weight-decay.
