  Multi-channel imaging data is a prevalent data format in scientific fields
such as astronomy and biology. The structured information and the high
dimensionality of these 3-D tensor data makes the analysis an intriguing but
challenging topic for statisticians and practitioners. The low-rank
scalar-on-tensor regression model, in particular, has received widespread
attention and has been re-formulated as a tensor Gaussian Process (Tensor-GP)
model with multi-linear kernel in Yu et al.(2018). In this paper, we extend the
Tensor-GP model by introducing an integrative dimensionality reduction
technique, called tensor contraction, with a Tensor-GP for a scalar-on-tensor
regression task with multi-channel imaging data. This is motivated by the solar
flare forecasting problem with high dimensional multi-channel imaging data. We
first estimate a latent, reduced-size tensor for each data tensor and then
apply a multi-linear Tensor-GP on the latent tensor data for prediction. We
introduce an anisotropic total-variation regularization when conducting the
tensor contraction to obtain a sparse and smooth latent tensor. We then propose
an alternating proximal gradient descent algorithm for estimation. We validate
our approach via extensive simulation studies and applying it to the solar
flare forecasting problem.
