In this work, we develop an optimization framework for problems whosesolutions are well-approximated by Hierarchical Tucker (HT) tensors, anefficient structured tensor format based on recursive subspace factorizations.By exploiting the smooth manifold structure of these tensors, we constructstandard optimization algorithms such as Steepest Descent and ConjugateGradient for completing tensors from missing entries. Our algorithmic frameworkis fast and scalable to large problem sizes as we do not require SVDs on theambient tensor space, as required by other methods. Moreover, we exploit thestructure of the Gramian matrices associated with the HT format to regularizeour problem, reducing overfitting for high subsampling ratios. We also findthat the organization of the tensor can have a major impact on completion fromrealistic seismic acquisition geometries. These samplings are far fromidealized randomized samplings that are usually considered in the literaturebut are realizable in practical scenarios. Using these algorithms, wesuccessfully interpolate large-scale seismic data sets and demonstrate thecompetitive computational scaling of our algorithms as the problem sizes grow.
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