Optimization on the Hierarchical Tucker manifold - applications to tensor completion

TL;DR

This paper introduces an efficient optimization framework on the Hierarchical Tucker tensor manifold for large-scale tensor completion, especially suited for realistic seismic data, avoiding costly SVDs and improving regularization.

Contribution

It develops scalable optimization algorithms on the HT manifold for tensor completion, incorporating structure-based regularization and handling realistic sampling geometries.

Findings

Algorithms successfully interpolate large seismic datasets.

Scalable methods outperform existing approaches in computational efficiency.

Regularization reduces overfitting in high subsampling scenarios.

Abstract

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.