  Tensor regression methods have been widely used to predict a scalar response
from covariates in the form of a multiway array. In many applications, the
regions of tensor covariates used for prediction are often spatially connected
with unknown shapes and discontinuous jumps on the boundaries. Moreover, the
relationship between the response and the tensor covariates can be nonlinear.
In this article, we develop a nonlinear Bayesian tensor additive regression
model to accommodate such spatial structure. A functional fused elastic net
prior is proposed over the additive component functions to comprehensively
model the nonlinearity and spatial smoothness, detect the discontinuous jumps,
and simultaneously identify the active regions. The great flexibility and
interpretability of the proposed method against the alternatives are
demonstrated by a simulation study and an analysis on facial feature data.
