@article {dasilva2014htuck,
	title = {Optimization on the Hierarchical Tucker manifold - applications to tensor completion},
	journal = {Linear Algebra and its Applications},
	volume = {481},
	year = {2015},
	note = {(Linear Algebra and its Applications)},
	month = {09},
	pages = {131-173},
	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.},
	keywords = {differential geometry, Gauss{\textendash}Newton, hierarchical tucker tensors, low-rank tensor, Riemannian manifold optimization, tensor completion},
	doi = {10.1016/j.laa.2015.04.015},
	url = {https://slim.gatech.edu/Publications/Public/Journals/LinearAlgebraAndItsApplications/2015/dasilva2014htuck/dasilva2014htuck.pdf},
	url2 = {http://www.sciencedirect.com/science/article/pii/S0024379515002530},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@conference {dasilva2013SAMPTAhtuck,
	title = {Hierarchical Tucker tensor optimization - applications to tensor completion},
	year = {2013},
	month = {07},
	abstract = {In this work, we develop an optimization framework for problems whose solutions are well-approximated by Hierarchical Tucker tensors, an efficient structured tensor format based on recursive subspace factorizations. Using the differential geometric tools presented here, we construct standard optimization algorithms such as Steepest Descent and Conjugate Gradient, for interpolating tensors in HT format. We also empirically examine the importance of one{\textquoteright}s choice of data organization in the success of tensor recovery by drawing upon insights from the Matrix Completion literature. Using these algorithms, we recover various seismic data sets with randomly missing source pairs.},
	keywords = {differential geometry, hierarchical tucker, riemannian optimization, SAMPTA, structured tensor, tensor interpolation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SAMPTA/2013/dasilva2013SAMPTAhtuck/dasilva2013SAMPTAhtuck.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SAMPTA/2013/dasilva2013SAMPTAhtuck/dasilva2013SAMPTAhtuck_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@conference {dasilva2013SEGhtuck,
	title = {Structured tensor missing-trace interpolation in the Hierarchical Tucker format},
	booktitle = {SEG Technical Program Expanded Abstracts},
	volume = {32},
	year = {2013},
	month = {09},
	pages = {3623-3627},
	publisher = {SEG},
	organization = {SEG},
	abstract = {Owing to the large scale and dimensionality of a 3D seismic experiment, acquiring fully-sampled data according to the Nyquist criterion is an exceedingly arduous and cost-prohibitive task. In this paper, we develop tools to interpolate 5D seismic volumes with randomly missing sources or receivers using a relatively novel tensor format known as the Hierarchical Tucker (HT) format. By exploiting the underlying smooth structure of HT tensors, specifically its smooth manifold structure, we develop solvers which are fast, immediately parallelizable, and SVD-free, making these solvers amenable to large-scale problems where SVD-based projection methods are far too costly. We also build on intuition of multidimensional sampling from the perspective of matrix-completion and demonstrate the ability of our algorithms to recover frequency slices even amidst very high levels of source subsampling on a synthetic large-scale 3D North Sea dataset.},
	keywords = {differential geometry, gauss newton, hierarchical tucker, riemannian optimization, SEG, structured tensor, tensor interpolation},
	doi = {10.1190/segam2013-0709.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2013/dasilva2013SEGhtuck/dasilva2013SEGhtuck.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2013/dasilva2013SEGhtuck/dasilva2013SEGhtuck_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
