Bayesian Adaptive Tucker Decompositions for Tensor Factorization

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.

相较于广泛应用的平行因子分析（PARAFAC）模型，塔克张量分解（Tucker tensor decomposition）对多路数据具备更优异的表征性能。然而其灵活性也带来了合理潜在多重秩选择的挑战。为解决预先指定潜在多重秩的痛点，本文提出一种贝叶斯自适应塔克分解模型，该模型通过无限递增收缩先验自动完成多重秩的推断。该模型在核心张量中引入局部稀疏性，可构建兼具丰富性与简约性的依赖结构。后验推断依托高效的自适应吉布斯采样（adaptive Gibbs sampler）实现，能够支持连续与二元两类数据，并在处理不完整多路数据时可便捷实现缺失数据插补。本文对所提出的建模框架的基本性质展开讨论，并给出理论佐证。仿真实验以及在化学计量学与复杂生态数据上的应用，均提供了该方法相较于现有张量分解方法具备显著优势的有力证据。