Statistical mechanics of low-rank tensor decomposition

TL;DR

This paper develops a theoretical framework for understanding low-rank tensor decomposition using Bayesian AMP algorithms, revealing phase transitions and outperforming traditional methods like ALS in noisy settings.

Contribution

It introduces Bayesian AMP algorithms for arbitrary tensor shapes and uses dynamic mean field theory to analyze their performance and phase transitions.

Findings

AMP outperforms ALS in noisy environments

Identifies phase transitions between inference regimes

Matches theoretical predictions with simulations

Abstract

Often, large, high dimensional datasets collected across multiple modalities can be organized as a higher order tensor. Low-rank tensor decomposition then arises as a powerful and widely used tool to discover simple low dimensional structures underlying such data. However, we currently lack a theoretical understanding of the algorithmic behavior of low-rank tensor decompositions. We derive Bayesian approximate message passing (AMP) algorithms for recovering arbitrarily shaped low-rank tensors buried within noise, and we employ dynamic mean field theory to precisely characterize their performance. Our theory reveals the existence of phase transitions between easy, hard and impossible inference regimes, and displays an excellent match with simulations. Moreover, it reveals several qualitative surprises compared to the behavior of symmetric, cubic tensor decomposition. Finally, we compare our AMP algorithm to the most commonly used algorithm, alternating least squares (ALS), and demonstrate that AMP significantly outperforms ALS in the presence of noise.