  With advances in data collection technologies, tensor data is assuming
increasing prominence in many applications and the problem of supervised tensor
learning has emerged as a topic of critical significance in the data mining and
machine learning community. Conventional methods for supervised tensor learning
mainly focus on learning kernels by flattening the tensor into vectors or
matrices, however structural information within the tensors will be lost. In
this paper, we introduce a new scheme to design structure-preserving kernels
for supervised tensor learning. Specifically, we demonstrate how to leverage
the naturally available structure within the tensorial representation to encode
prior knowledge in the kernel. We proposed a tensor kernel that can preserve
tensor structures based upon dual-tensorial mapping. The dual-tensorial mapping
function can map each tensor instance in the input space to another tensor in
the feature space while preserving the tensorial structure. Theoretically, our
approach is an extension of the conventional kernels in the vector space to
tensor space. We applied our novel kernel in conjunction with SVM to real-world
tensor classification problems including brain fMRI classification for three
different diseases (i.e., Alzheimer's disease, ADHD and brain damage by HIV).
Extensive empirical studies demonstrate that our proposed approach can
effectively boost tensor classification performances, particularly with small
sample sizes.
