Tensor decompositions play a crucial role in numerous applications related to multi-way data analysis. By employing a Bayesian framework with sparsity-inducing priors, Bayesian Tensor Ring (BTR) factorization offers probabilistic estimates and an effective approach for automatically adapting the tensor ring rank during the learning process. However, previous BTR method employs an Automatic Relevance Determination (ARD) prior, which can lead to sub-optimal solutions. Besides, it solely focuses on continuous data, whereas many applications involve discrete data. More importantly, it relies on the Coordinate-Ascent Variational Inference (CAVI) algorithm, which is inadequate for handling large tensors with extensive observations. These limitations greatly limit its application scales and scopes, making it suitable only for small-scale problems, such as image/video completion. To address these issues, we propose a novel BTR model that incorporates a nonparametric Multiplicative Gamma Process (MGP) prior, known for its superior accuracy in identifying latent structures. To handle discrete data, we introduce the P\'olya-Gamma augmentation for closed-form updates. Furthermore, we develop an efficient Gibbs sampler for consistent posterior simulation, which reduces the computational complexity of previous VI algorithm by two orders, and an online EM algorithm that is scalable to extremely large tensors. To showcase the advantages of our model, we conduct extensive experiments on both simulation data and real-world applications.

翻译：张量分解在多路数据分析的众多应用中扮演着关键角色。通过采用具有稀疏性诱导先验的贝叶斯框架，贝叶斯张量环（BTR）分解提供了概率估计，并为学习过程中自动调整张量环秩提供了一种有效方法。然而，以往的BTR方法采用自动相关性确定（ARD）先验，可能导致次优解。此外，该方法仅关注连续数据，而许多应用涉及离散数据。更重要的是，它依赖于坐标上升变分推断（CAVI）算法，该算法不足以处理具有大量观测值的大规模张量。这些局限性极大地限制了其应用规模和范围，使其仅适用于小规模问题，如图像/视频补全。为解决这些问题，我们提出了一种新颖的BTR模型，该模型结合了非参数乘法伽马过程（MGP）先验，该先验在识别潜在结构方面以更高的准确性而闻名。为处理离散数据，我们引入了Pólya-Gamma增强以实现闭式更新。此外，我们开发了一种高效的吉布斯采样器用于一致性后验模拟，将先前VI算法的计算复杂度降低了两个数量级，并提出了一种可扩展至极大张量的在线EM算法。为展示我们模型的优势，我们在仿真数据和实际应用上进行了大量实验。