Large-Scale Bayesian Tensor Reconstruction: An Approximate Message Passing Solution

TL;DR

This paper introduces CP-GAMP, a scalable Bayesian tensor decomposition algorithm that efficiently reconstructs large tensors with limited observations by avoiding matrix inversions and jointly estimating tensor rank and noise.

Contribution

The paper presents CP-GAMP, a novel scalable Bayesian CPD method using GAMP and EM routines, significantly improving efficiency for large-scale tensor reconstruction.

Findings

Reduces runtime by 82.7% compared to existing methods

Maintains comparable reconstruction accuracy on synthetic tensors

Efficiently handles tensors with only 20% observed elements

Abstract

Tensor CANDECOMP/PARAFAC decomposition (CPD) is a fundamental model for tensor reconstruction. Although the Bayesian framework allows for principled uncertainty quantification and automatic hyperparameter learning, existing methods do not scale well for large tensors because of high-dimensional matrix inversions. To this end, we introduce CP-GAMP, a scalable Bayesian CPD algorithm. This algorithm leverages generalized approximate message passing (GAMP) to avoid matrix inversions and incorporates an expectation-maximization routine to jointly infer the tensor rank and noise power. Through multiple experiments, for synthetic 100x100x100 rank 20 tensors with only 20% elements observed, the proposed algorithm reduces runtime by 82.7% compared to the state-of-the-art variational Bayesian CPD method, while maintaining comparable reconstruction accuracy.