  Sparse modeling for signal processing and machine learning has been at the
focus of scientific research for over two decades. Among others, supervised
sparsity-aware learning comprises two major paths paved by: a) discriminative
methods and b) generative methods. The latter, more widely known as Bayesian
methods, enable uncertainty evaluation w.r.t. the performed predictions.
Furthermore, they can better exploit related prior information and naturally
introduce robustness into the model, due to their unique capacity to
marginalize out uncertainties related to the parameter estimates. Moreover,
hyper-parameters associated with the adopted priors can be learnt via the
training data. To implement sparsity-aware learning, the crucial point lies in
the choice of the function regularizer for discriminative methods and the
choice of the prior distribution for Bayesian learning. Over the last decade or
so, due to the intense research on deep learning, emphasis has been put on
discriminative techniques. However, a come back of Bayesian methods is taking
place that sheds new light on the design of deep neural networks, which also
establish firm links with Bayesian models and inspire new paths for
unsupervised learning, such as Bayesian tensor decomposition.
  The goal of this article is two-fold. First, to review, in a unified way,
some recent advances in incorporating sparsity-promoting priors into three
highly popular data modeling tools, namely deep neural networks, Gaussian
processes, and tensor decomposition. Second, to review their associated
inference techniques from different aspects, including: evidence maximization
via optimization and variational inference methods. Challenges such as small
data dilemma, automatic model structure search, and natural prediction
uncertainty evaluation are also discussed. Typical signal processing and
machine learning tasks are demonstrated.
