  Inverse problems in image and audio, and super-resolution in particular, can
be seen as high-dimensional structured prediction problems, where the goal is
to characterize the conditional distribution of a high-resolution output given
its low-resolution corrupted observation. When the scaling ratio is small,
point estimates achieve impressive performance, but soon they suffer from the
regression-to-the-mean problem, result of their inability to capture the
multi-modality of this conditional distribution. Modeling high-dimensional
image and audio distributions is a hard task, requiring both the ability to
model complex geometrical structures and textured regions. In this paper, we
propose to use as conditional model a Gibbs distribution, where its sufficient
statistics are given by deep convolutional neural networks. The features
computed by the network are stable to local deformation, and have reduced
variance when the input is a stationary texture. These properties imply that
the resulting sufficient statistics minimize the uncertainty of the target
signals given the degraded observations, while being highly informative. The
filters of the CNN are initialized by multiscale complex wavelets, and then we
propose an algorithm to fine-tune them by estimating the gradient of the
conditional log-likelihood, which bears some similarities with Generative
Adversarial Networks. We evaluate experimentally the proposed approach in the
image super-resolution task, but the approach is general and could be used in
other challenging ill-posed problems such as audio bandwidth extension.
