Riemannian Preconditioned Coordinate Descent for Low Multilinear Rank Approximation

.This paper presents a memory-efficient, first-order method for low multilinear 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.KeywordsTucker decompositionRiemannian optimizationpreconditioningcoordinate descentRiemannian metricMSC codes15A6949M3753A4565F08