  This paper presents a memory efficient, first-order method for low
multi-linear rank approximation of high-order, high-dimensional tensors. In our
method, we exploit the second-order information of the cost function and the
constraints to suggest a new Riemannian metric on the Grassmann manifold. We
use a Riemmanian coordinate descent method for solving the problem, and also
provide a global convergence analysis matching that of the coordinate descent
method in the Euclidean setting. We also show that each step of our method with
the unit step-size is actually a step of the orthogonal iteration algorithm.
Experimental results show the computational advantage of our method for
high-dimensional tensors.
