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Factorizing low-rank matrices is a problem with many applications in machinelearning and statistics, ranging from sparse PCA to community detection andsub-matrix localization. For probabilistic models in the Bayes optimal setting,general expressions for the mutual information have been proposed usingpowerful heuristic statistical physics computations via the replica and cavitymethods, and proven in few specific cases by a variety of methods. Here, we usethe spatial coupling methodology developed in the framework of error correctingcodes, to rigorously derive the mutual information for the symmetric rank-onecase. We characterize the detectability phase transitions in a large set ofestimation problems, where we show that there exists a gap between whatcurrently known polynomial algorithms (in particular spectral methods andapproximate message-passing) can do and what is expected informationtheoretically. Moreover, we show that the computational gap vanishes for theproposed spatially coupled model, a promising feature with many possibleapplications. Our proof technique has an interest on its own and exploits threeessential ingredients: the interpolation method first introduced in statisticalphysics, the analysis of approximate message-passing algorithms firstintroduced in compressive sensing, and the theory of threshold saturation forspatially coupled systems first developed in coding theory. Our approach isvery generic and can be applied to many other open problems in statisticalestimation where heuristic statistical physics predictions are available.

Rank-one matrix estimation: analysis of algorithmic and information theoretic limits by the spatial coupling method