Multiple Graph Structure Learning: a comparative analysis

In the context of analysing multivariate Gaussian distributions under different experimental conditions, recent studies have focused on retrieving the patterns of the conditional independences between pairs of variables for each condition. Given the representation of non-zero partial correlations as edges in a graph, we refer to this domain as Multiple Graph Structure Learning. In application problems that assume some similarity between the graph structures, it has been suggested in the literature that learning the graphs jointly would be advantageous with respect to learning them separately. As an alternative, the graphs can be learnt directly from the difference of the concentration matrices. The aim of this thesis is to understand the advantages and limitations of such learning methods. In order to do so, we compare these strategies by constructing a comprehensive and detailed simulation study analysis that includes different graph structures, different sample sizes, different dimensions and different levels of similarity between the experimental conditions. We evaluate the performance of the methods using the precision and recall indexes. From the results of our simulation, it is evident that the underlying limitation of all the graph structure learning methods resides in the model selection, which corresponds to the choice of l1-norm penalty terms. This leads to the identification of graphs with highly variable densities, which hinders the method comparison. We then impose that the models reproduce the true graph densities and we explore how different the resulting graphs are with respect to each learning method and simulation scenario.