  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}.
