Tucker tensor decomposition offers a more effective representation for multiway data compared to the widely used PARAFAC model. However, its flexibility brings the challenge of selecting the appropriate latent multi-rank. To overcome the issue of pre-selecting the latent multi-rank, we introduce a Bayesian adaptive Tucker decomposition model that infers the multi-rank automatically via an infinite increasing shrinkage prior. The model introduces local sparsity in the core tensor, inducing rich and at the same time parsimonious dependency structures. Posterior inference proceeds via an efficient adaptive Gibbs sampler, supporting both continuous and binary data and allowing for straightforward missing data imputation when dealing with incomplete multiway data. We discuss fundamental properties of the proposed modeling framework, providing theoretical justification. Simulation studies and applications to chemometrics and complex ecological data offer compelling evidence of its advantages over existing tensor factorization methods.

翻译：相较于广泛使用的PARAFAC模型，Tucker张量分解为多维数据提供了更有效的表示方法。然而，其灵活性带来了选择合适潜在多重秩的挑战。为克服预先设定潜在多重秩的问题，我们提出了一种贝叶斯自适应Tucker分解模型，该模型通过无限递增收缩先验自动推断多重秩。该模型在核心张量中引入局部稀疏性，从而诱导出丰富且简约的依赖结构。后验推断通过高效的自适应吉布斯采样器进行，支持连续型和二值型数据，并能对不完整多维数据直接进行缺失值插补。我们讨论了所提建模框架的基本性质，并提供了理论依据。模拟研究及在化学计量学和复杂生态数据中的应用，有力证明了该方法相较于现有张量因子化方法的优势。