# Large-Scale Bayesian Tensor Reconstruction via Approximate Message Passing

arXiv: https://arxiv.org/abs/2505.16305 · HTML: https://mlsanity.com/paper/2505.16305/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 9 of 9)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Nonparametric Iterative Machine Teaching](https://mlsanity.com/paper/2306.03007/) (2023)
- [Nonparametric Teaching for Graph Property Learners](https://mlsanity.com/paper/2505.14170/) (2025)
- [Statistical mechanics of low-rank tensor decomposition](https://mlsanity.com/paper/1810.10065/) (2018)
- [Efficient Nonparametric Tensor Decomposition for Binary and Count Data](https://mlsanity.com/paper/2401.07711/) (2024)
- [Scalable Bayesian Tensor Ring Factorization for Multiway Data Analysis](https://mlsanity.com/paper/2412.03321/) (2024)
- [Variational Bayesian inference for CP tensor completion with side information](https://mlsanity.com/paper/2206.12486/) (2022)

## Abstract (arXiv)

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.