Large-Scale Bayesian Tensor Reconstruction via Approximate Message Passing

Abstract: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.

Accepted at IEEE ICDM 2026. Code: this https URL

Machine Learning (cs.LG); Signal Processing (eess.SP)

arXiv:2505.16305 [cs.LG]

(or arXiv:2505.16305v3 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2505.16305 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.