  With the advances in data acquisition technology, tensor objects are
collected in a variety of applications including multimedia, medical and
hyperspectral imaging. As the dimensionality of tensor objects is usually very
high, dimensionality reduction is an important problem. Most of the current
tensor dimensionality reduction methods rely on finding low-rank linear
representations using different generative models. However, it is well-known
that high-dimensional data often reside in a low-dimensional manifold.
Therefore, it is important to find a compact representation, which uncovers the
low dimensional tensor structure while respecting the intrinsic geometry. In
this paper, we propose a graph regularized tensor train (GRTT) decomposition
that learns a low-rank tensor train model that preserves the local
relationships between tensor samples. The proposed method is formulated as a
nonconvex optimization problem on the Stiefel manifold and an efficient
algorithm is proposed to solve it. The proposed method is compared to existing
tensor based dimensionality reduction methods as well as tensor manifold
embedding methods for unsupervised learning applications.
