Convergence analysis of an SVD-based algorithm for the best rank-1 tensor approximation

This paper revisits the classical problem of finding the best rank-1 approximation to a generic tensor. The main focus is on providing a mathematical proof for the convergence of the iterates of an SVD-based algorithm. In contrast to the conventional approach by the so called alternating least squares (ALS) method that works to adjust one factor a time, the SVD-based algorithms improve two factors simultaneously. The ALS method is easy to implement, but suffers from slow convergence and easy stagnation at a local solution. It has been suggested recently that the SVD-algorithm might have a better limiting behavior leading to better approximations, yet a theory of convergence has been elusive in the literature. This note proposes a simple tactic to partially close that gap.