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In this paper, we consider the estimation of a low Tucker rank tensor from anumber of noisy linear measurements. The general problem covers many specificexamples arising from applications, including tensor regression, tensorcompletion, and tensor PCA/SVD. We consider an efficient RiemannianGauss-Newton (RGN) method for low Tucker rank tensor estimation. Different fromthe generic (super)linear convergence guarantee of RGN in the literature, weprove the first local quadratic convergence guarantee of RGN for low-ranktensor estimation in the noisy setting under some regularity conditions andprovide the corresponding estimation error upper bounds. A deterministicestimation error lower bound, which matches the upper bound, is provided thatdemonstrates the statistical optimality of RGN. The merit of RGN is illustratedthrough two machine learning applications: tensor regression and tensor SVD.Finally, we provide the simulation results to corroborate our theoreticalfindings.

Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence