  Factorizing low-rank matrices is a problem with many applications in machine
learning and statistics, ranging from sparse PCA to community detection and
sub-matrix localization. For probabilistic models in the Bayes optimal setting,
general expressions for the mutual information have been proposed using
powerful heuristic statistical physics computations via the replica and cavity
methods, and proven in few specific cases by a variety of methods. Here, we use
the spatial coupling methodology developed in the framework of error correcting
codes, to rigorously derive the mutual information for the symmetric rank-one
case. We characterize the detectability phase transitions in a large set of
estimation problems, where we show that there exists a gap between what
currently known polynomial algorithms (in particular spectral methods and
approximate message-passing) can do and what is expected information
theoretically. Moreover, we show that the computational gap vanishes for the
proposed spatially coupled model, a promising feature with many possible
applications. Our proof technique has an interest on its own and exploits three
essential ingredients: the interpolation method first introduced in statistical
physics, the analysis of approximate message-passing algorithms first
introduced in compressive sensing, and the theory of threshold saturation for
spatially coupled systems first developed in coding theory. Our approach is
very generic and can be applied to many other open problems in statistical
estimation where heuristic statistical physics predictions are available.
