# Statistical and computational phase transitions in spiked tensor estimation

## TL;DR

This paper rigorously analyzes phase transitions in spiked tensor estimation, showing when Bayesian methods and AMP algorithms succeed or fail, revealing a wider 'easy' region than previously known.

## Contribution

It provides a rigorous computation of mutual information and MMSE, and characterizes the algorithmic and information-theoretic phase transitions in tensor factorization.

## Key findings

- AMP achieves MMSE in a large parameter region
- Existence of a 'hard' region where AMP fails
- Factorization is algorithmically 'easy' over a wider range

## 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.