Functional Bayesian Tucker Decomposition for Continuous-indexed Tensor Data
This work addresses a domain-specific problem for researchers and practitioners dealing with continuous-indexed data in fields like geospatial analysis, offering a novel method but with incremental improvements over existing tensor models.
The authors tackled the problem of applying Tucker decomposition to continuous-indexed tensor data, such as geographic coordinates, by proposing Functional Bayesian Tucker Decomposition (FunBaT), which uses Gaussian processes and stochastic differential equations to model latent functions and achieve scalable inference, demonstrating advantages in synthetic and real-world applications.
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}.