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 were
finite objects in each aspect or mode, corresponding to discrete indexes of
data entries. However, many real-world data are not naturally posed in the
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, and
then convert the GPs into a state-space prior by constructing an equivalent
stochastic differential equation (SDE) to reduce computational cost. An
efficient inference algorithm is further 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.

Tucker 分解是一种处理多方面数据的强大张量模型，本文提出了将其扩展到连续索引数据的功能性贝叶斯 Tucker 分解方法，使用高斯过程作为函数先验，并采用消息传递技术进行可扩展的后验近似计算。