A NEWTON-GRASSMANN METHOD FOR COMPUTING THE BEST MULTILINEAR RANK-(r1, r2, r3) APPROXIMATION OF A TENSOR

We derive a Newton method for computing the best rank-(r(1), r(2), r(3)) approximation of a given J x K x L tensor A. The problem is formulated as an approximation problem on a product of Grassmann manifolds. Incorporating the manifold structure into Newton's method ensures that all iterates generated by the algorithm are points on the Grassmann manifolds. We also introduce a consistent notation for matricizing a tensor, for contracted tensor products and some tensor-algebraic manipulations, which simplify the derivation of the Newton equations and enable straightforward algorithmic implementation. Experiments show a quadratic convergence rate for the Newton-Grassmann algorithm.