  Low rank orthogonal tensor approximation (LROTA) is an important problem in
tensor computations and their applications. A classical and widely used
algorithm is the alternating polar decomposition method (APD). In this article,
an improved version iAPD of the classical APD is proposed. For the first time,
all the following four fundamental properties are established for iAPD: (i) the
algorithm converges globally and the whole sequence converges to a KKT point
without any assumption; (ii) it exhibits an overall sublinear convergence with
an explicit rate which is sharper than the usual $O(1/k)$ for first order
methods in optimization; (iii) more importantly, it converges $R$-linearly for
a generic tensor without any assumption; (iv) for almost all LROTA problems,
iAPD reduces to APD after finitely many iterations if it converges to a local
minimizer.
