Low-rank tensor decomposition (TD) is usually effective on clean, fully observed data, but it often degrades under severe missingness or noise. Low-rankness is itself a useful but limited structural prior, and additional handcrafted priors (e.g., sparsity or smoothness) still fall short of capturing the rich statistics of real-world data. To compensate for this weak inductive bias under heavy corruption, one would like to inject a learned, data-driven prior; however, the state-of-the-art diffusion models are not readily compatible with current TD and tractable posterior inference. To address these challenges, we introduce DiffBCP, a hybrid-prior Bayesian CP decomposition framework that couples a cumulative shrinkage process prior over the CP factors for automatic rank selection with an off-the-shelf pre-trained diffusion model as an implicit data prior on the reconstructed tensor. To make posterior inference tractable despite the coupling among the likelihood, low-rank constraint, and diffusion prior, we develop a split Gibbs sampler: CP factors admit conjugate updates, while the diffusion block is sampled via low-rank-guided denoising. A noise-adaptive coupling schedule further reduces sensitivity to hand-tuned annealing. Experiments on image inpainting and denoising, including high-resolution out-of-distribution images, show consistent gains over Bayesian, nonlinear, and plug-and-play TD baselines.

翻译：摘要：低秩张量分解（TD）通常对干净、完全观测的数据有效，但在严重缺失或噪声条件下性能往往下降。低秩性本身是一种有用但有限的先验结构，而额外的手工先验（如稀疏性或平滑性）仍难以捕捉真实世界数据的丰富统计特征。为弥补这种重度污染下的弱归纳偏差，需要引入学习所得的、数据驱动的先验；然而，当前最先进的扩散模型难以与现有TD及可处理的后验推理兼容。为解决这些挑战，我们提出DiffBCP——一种混合先验的贝叶斯CP分解框架，该框架将CP因子上的累积收缩过程先验（用于自动秩选择）与现成的预训练扩散模型（作为重建张量上的隐式数据先验）相耦合。为使似然函数、低秩约束和扩散先验耦合下的后验推理可处理，我们开发了分裂吉布斯采样器：CP因子允许共轭更新，而扩散块通过低秩引导的去噪过程进行采样。噪声自适应耦合调度进一步降低了对手工调参退火的敏感性。图像修复与去噪实验（包括高分辨率分布外图像）表明，本方法在贝叶斯、非线性及即插即用TD基线上持续取得性能提升。