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-(r₁,r₂,r₃) approximation of a given J× K× 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.