Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at \url{https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition}.

翻译：Tucker分解是一种处理多维度数据的强大张量模型。该模型通过将网格结构数据分解为核心张量与一组对象表示（因子）之间的相互作用，揭示了数据的低秩特性。这种分解的一个基本假设是每个维度或模态中存在有限的对象，对应数据条目的离散索引。然而，现实世界的数据往往并不天然符合这一设定。例如，地理数据以经纬度坐标的连续索引形式表示，无法直接适配张量模型。为将Tucker分解推广至此类场景，我们提出了函数贝叶斯Tucker分解（FunBaT）。我们将连续索引数据视为Tucker核心与一组潜在函数之间的相互作用，采用高斯过程（GP）作为函数先验来建模潜在函数。进一步地，通过构建等效的随机微分方程（SDE），我们将每个GP转化为状态空间先验以降低计算成本。基于先进的置信传播技术，我们开发了一种高效的推理算法，用于可扩展的后验近似。在合成数据及多个实际应用中的实验表明，我们的方法具有显著优势。FunBaT的代码已开源发布于\url{https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition}。