Large-Scale Bayesian Tensor Reconstruction via Approximate Message Passing

While CANDECOMP/PARAFAC (CP) decomposition (CPD) is fundamental for tensor reconstruction, Bayesian CPD often scales poorly because variational updates require repeated matrix inversions. We develop CP generalized approximate message passing (CP-GAMP) for incomplete noisy Bayesian CPD. The algorithm uses Gaussian message approximations to avoid high-dimensional inversions, and it combines a Bernoulli-Gaussian prior with expectation-maximization updates to estimate effective CP rank and noise variance. We also give a formal state evolution (SE) recursion and relate its fixed points to replica-symmetric saddle points, so CP-GAMP's SE-predicted error can be compared with the formal replica-symmetric minimum mean-squared error (MMSE) benchmark in the matched limit. Synthetic and image-inpainting experiments show that CP-GAMP substantially reduces runtime relative to variational Bayesian CPD while maintaining competitive reconstruction accuracy.