  In this paper, we consider the estimation of a low Tucker rank tensor from a
number of noisy linear measurements. The general problem covers many specific
examples arising from applications, including tensor regression, tensor
completion, and tensor PCA/SVD. We consider an efficient Riemannian
Gauss-Newton (RGN) method for low Tucker rank tensor estimation. Different from
the generic (super)linear convergence guarantee of RGN in the literature, we
prove the first local quadratic convergence guarantee of RGN for low-rank
tensor estimation in the noisy setting under some regularity conditions and
provide the corresponding estimation error upper bounds. A deterministic
estimation error lower bound, which matches the upper bound, is provided that
demonstrates the statistical optimality of RGN. The merit of RGN is illustrated
through two machine learning applications: tensor regression and tensor SVD.
Finally, we provide the simulation results to corroborate our theoretical
findings.
