In this paper, we introduce two Gromov-Wasserstein-type distances on the set of Gaussian mixture models. The first one takes the form of a Gromov-Wasserstein distance between two discrete distributionson the space of Gaussian measures. This distance can be used as an alternative to Gromov-Wasserstein for applications which only require to evaluate how far the distributions are from each other but does not allow to derive directly an optimal transportation plan between clouds of points. To design a way to define such a transportation plan, we introduce another distance between measures living in incomparable spaces that turns out to be closely related to Gromov-Wasserstein. When restricting the set of admissible transportation couplings to be themselves Gaussian mixture models in this latter, this defines another distance between Gaussian mixture models that can be used as another alternative to Gromov-Wasserstein and which allows to derive an optimal assignment between points. Finally, we design a transportation plan associated with the first distance by analogy with the second, and we illustrate their practical uses on medium-to-large scale problems such as shape matching and hyperspectral image color transfer.

Community detection is one of the most critical problems in modern network science. Its applications can be found in various fields, from protein modeling to social network analysis. Recently, many papers appeared studying the problem of overlapping community detection, where each node of a network may belong to several communities. In this work, we consider Mixed-Membership Stochastic Block Model (MMSB) first proposed by Airoldi et al. (2008). MMSB provides quite a general setting for modeling overlapping community structure in graphs. The central question of this paper is to reconstruct relations between communities given an observed network. We compare different approaches and establish the minimax lower bound on the estimation error. Then, we propose a new estimator that matches this lower bound. Theoretical results are proved under fairly general conditions on the considered model. Finally, we illustrate the theory in a series of experiments.