  We investigate the sample size requirement for exact recovery of a high order
tensor of low rank from a subset of its entries. In the Tucker decomposition
framework, we show that the Riemannian optimization algorithm with initial
value obtained from a spectral method can reconstruct a tensor of size $n\times
n \times\cdots \times n$ tensor of ranks $(r,\cdots,r)$ with high probability
from as few as $O((r^d+dnr)\log(d))$ entries. In the case of order 3 tensor,
the entries can be asymptotically as few as $O(nr)$ for a low rank large
tensor. We show the theoretical guarantee condition for the recovery. The
analysis relies on the tensor restricted isometry property (tensor RIP) and the
curvature of the low rank tensor manifold. Our algorithm is computationally
efficient and easy to implement. Numerical results verify that the algorithms
are able to recover a low rank tensor from minimum number of measurements. The
experiments on hyperspectral images recovery also show that our algorithm is
capable of real world signal processing problems.
