  We study extensions of compressive sensing and low rank matrix recovery
(matrix completion) to the recovery of low rank tensors of higher order from a
small number of linear measurements. While the theoretical understanding of low
rank matrix recovery is already well-developed, only few contributions on the
low rank tensor recovery problem are available so far. In this paper, we
introduce versions of the iterative hard thresholding algorithm for several
tensor decompositions, namely the higher order singular value decomposition
(HOSVD), the tensor train format (TT), and the general hierarchical Tucker
decomposition (HT). We provide a partial convergence result for these
algorithms which is based on a variant of the restricted isometry property of
the measurement operator adapted to the tensor decomposition at hand that
induces a corresponding notion of tensor rank. We show that subgaussian
measurement ensembles satisfy the tensor restricted isometry property with high
probability under a certain almost optimal bound on the number of measurements
which depends on the corresponding tensor format. These bounds are extended to
partial Fourier maps combined with random sign flips of the tensor entries.
Finally, we illustrate the performance of iterative hard thresholding methods
for tensor recovery via numerical experiments where we consider recovery from
Gaussian random measurements, tensor completion (recovery of missing entries),
and Fourier measurements for third order tensors.
