Graphical Markov Models (GMM)
A central aspect of statistical science is the assessment of dependence among stochastic variables. The familiar concepts of correlation, regression, and prediction are special cases, and identification of causal relationships ultimately rests on representations of multivariate dependence. Graphical Markov models (GMM) use graphs, either undirected, directed, or mixed, to represent multivariate dependences in a visual and computationally efficient manner. A GMM is usually constructed by specifying local dependences for each variable, equivalently, node of the graph in terms of its immediate neighbors and/or parents by means of undirected and/or directed edges. This simple local specification can represent a highly varied and complex system of multivariate dependences by means of the global structure of the graph, thereby obtaining efficiency in modeling, inference, and probabilistic calculations. For a fixed graph, equivalently model, the classical methods of statistical inference may be utilized. In many applied domains, however, such as expert systems for medical diagnosis or weather forecasting, or the analysis of gene-expression data, the graph is unknown and is itself the first goal of the analysis. This poses numerous challenges, including the following:
• The numbers of possible graphs and models grow superexponentially in the number of variables.
• Distinct graphs G may be Markov equivalent = statistically indistinguishable.
• Conversely, the same graph may possess different Markov interpretations. …

Convolutional Dictionary Learning
Convolutional sparse representations are a form of sparse representation with a dictionary that has a structure that is equivalent to convolution with a set of linear filters. While effective algorithms have recently been developed for the convolutional sparse coding problem, the corresponding dictionary learning problem is substantially more challenging. Furthermore, although a number of different approaches have been proposed, the absence of thorough comparisons between them makes it difficult to determine which of them represents the current state of the art. The present work both addresses this deficiency and proposes some new approaches that outperform existing ones in certain contexts. A thorough set of performance comparisons indicates a very wide range of performance differences among the existing and proposed methods, and clearly identifies those that are the most effective. …