This paper considers probabilistic estimation of a low-rank matrix from non-linear element-wise measurements of its elements. We derive the corresponding approximate message passing (AMP) algorithm and its state evolution. Relying on non-rigorous but standard assumptions motivated by statistical physics, we characterize the minimum mean squared error (MMSE) achievable information theoretically and with the AMP algorithm. Unlike in related problems of linear estimation, in the present setting the MMSE depends on the output channel only trough a single parameter - its Fisher information. We illustrate this striking finding by analysis of submatrix localization, and of detection of communities hidden in a dense stochastic block model. For this example we locate the computational and statistical boundaries that are not equal for rank larger than four.
Our results for the Bayes-optimal estimation error of community detection in dense stochastic block model are of independent interest. Analogous results were derived for the sparse case in @cite_15 . In the dense case only MSE-suboptimal spectral methods were evaluated @cite_25 . We also unveil a hard phase existing in this problem for rank @math , and becoming very wide for @math .
@cite_15 @cite_25
2004531067 2014259951
In this paper we extend our previous work on the stochastic block model, a commonly used generative model for social and biological networks, and the problem of inferring functional groups or communities from the topology of the network. We use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram. We describe in detail properties of the detectability undetectability phase transition and the easy hard phase transition for the community detection problem. Our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way, i.e., that maximizes the overlap with the underlying group memberships, and learning the underlying parameters of the block model. Finally, we apply the algorithm to two examples of real-world networks and discuss its performance. We study networks that display community structure -- groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence demonstrate the presence of a phase transition in matrix methods for community detection, such as the popular modularity maximization method. The transition separates a regime in which such methods successfully detect the community structure from one in which the structure is present but is not detected. By comparing these results with recent analyses of maximum-likelihood methods we are able to show that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails.