Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula

@inproceedings{Barbier2016MutualIF,
  title={Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula},
  author={Jean Barbier and Mohamad Dia and Nicolas Macris and Florent Krzakala and Thibault Lesieur and Lenka Zdeborov{\'a}},
  booktitle={NIPS},
  year={2016}
}
Factorizing low-rank matrices has many applications in machine learning and statistics. For probabilistic models in the Bayes optimal setting, a general expression for the mutual information has been proposed using heuristic statistical physics computations, and proven in few specific cases. Here, we show how to rigorously prove the conjectured formula for the symmetric rank-one case. This allows to express the minimal mean-square-error and to characterize the detectability phase transitions in… CONTINUE READING
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