Riemannian preconditioned coordinate descent for low multi-linear rank approximation

This paper presents a fast, memory efficient, optimization-based, 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 local convergence analysis matching that of the coordinate descent method in the Euclidean setting. We also show that each step of our method with 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.