提出快速贝叶斯张量分解方法，显著提速且精度不降。
Large-Scale Bayesian Tensor Reconstruction via Approximate Message Passing
- 用消息传递近似避免高维矩阵求逆，提升计算效率。
- 在合成数据和图像修复任务中，运行时间大幅减少，精度仍具竞争力。
- 理论分析与最小均方误差基准对齐，适用于高维张量重建场景。
虽然CANDECOMP/PARAFAC (CP)分解是张量重构的基础，但贝叶斯CPD通常因变分更新需反复进行矩阵求逆而难以扩展。本文提出用于不完备噪声数据的贝叶斯CPD的广义近似消息传递算法（CP-GAMP）。该方法采用高斯消息近似避免高维求逆，并结合伯努利-高斯先验与期望最大化更新，估计有效CP秩与噪声方差。我们还给出了形式化状态演化（SE）递推关系，并将其不动点与复制对称鞍点关联，使CP-GAMP预测的误差可与匹配极限下的复制对称最小均方误差（MMSE）基准比较。合成数据与图像修复实验表明，相较于变分贝叶斯CPD，CP-GAMP显著降低运行时间，同时保持相当的重构精度。
原文摘要 · Abstract (English)
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.
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