Statistical and computational phase transitions in spiked tensor estimation

Thibault Lesieur; Léo Miolane; Marc Lelarge; Florent Krzakala; Lenka Zdeborová

doi:10.1109/isit.2017.8006580

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arXiv · 1701.08010

Statistical and computational phase transitions in spiked tensor estimation

Thibault Lesieur·Léo Miolane·Marc Lelarge·Florent Krzakala·Lenka Zdeborová

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Abstract

We consider tensor factorizations using a generative model and a Bayesian approach. We compute rigorously the mutual information, the Minimal Mean Squared Error (MMSE), and unveil information-theoretic phase transitions. In addition, we study the performance of Approximate Message Passing (AMP) and show that it achieves the MMSE for a large set of parameters, and that factorization is algorithmically "easy" in a much wider region than previously believed. It exists, however, a "hard" region where AMP fails to reach the MMSE and we conjecture that no polynomial algorithm will improve on AMP.

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Thibault Lesieur, Léo Miolane, Marc Lelarge, Florent Krzakala, Lenka Zdeborová. 2017-01-27. Statistical and computational phase transitions in spiked tensor estimation. https://doi.org/10.1109/isit.2017.8006580

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