  In this work, we develop an optimization framework for problems whose
solutions are well-approximated by Hierarchical Tucker (HT) tensors, an
efficient structured tensor format based on recursive subspace factorizations.
By exploiting the smooth manifold structure of these tensors, we construct
standard optimization algorithms such as Steepest Descent and Conjugate
Gradient for completing tensors from missing entries. Our algorithmic framework
is fast and scalable to large problem sizes as we do not require SVDs on the
ambient tensor space, as required by other methods. Moreover, we exploit the
structure of the Gramian matrices associated with the HT format to regularize
our problem, reducing overfitting for high subsampling ratios. We also find
that the organization of the tensor can have a major impact on completion from
realistic seismic acquisition geometries. These samplings are far from
idealized randomized samplings that are usually considered in the literature
but are realizable in practical scenarios. Using these algorithms, we
successfully interpolate large-scale seismic data sets and demonstrate the
competitive computational scaling of our algorithms as the problem sizes grow.
