@conference {dasilva2013EAGEhtucktensor,
	title = {Hierarchical Tucker tensor optimization - applications to 4D seismic data interpolation},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2013},
	month = {06},
	abstract = {In this work, we develop optimization algorithms on the manifold of Hierarchical Tucker (HT) tensors, an extremely efficient format for representing high-dimensional tensors exhibiting particular low-rank structure. With some minor alterations to existing theoretical developments, we develop an optimization framework based on the geometric understanding of HT tensors as a smooth manifold, a generalization of smooth curves/surfaces. Building on the existing research of solving optimization problems on smooth manifolds, we develop Steepest Descent and Conjugate Gradient methods for HT tensors. The resulting algorithms converge quickly, are immediately parallelizable, and do not require the computation of SVDs. We also extend ideas about favourable sampling conditions for missing-data recovery from the field of Matrix Completion to Tensor Completion and demonstrate how the organization of data can affect the success of recovery. As a result, if one has data with randomly missing source pairs, using these ideas, coupled with an efficient solver, one can interpolate large-scale seismic data volumes with missing sources and/or receivers by exploiting the multidimensional dependencies in the data. We are able to recover data volumes amidst extremely high subsampling ratios (in some cases, > 75\%) using this approach.},
	keywords = {3D data interpolation, EAGE, riemannian optimization, structured tensor},
	doi = {10.3997/2214-4609.20130390},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2013/dasilva2013EAGEhtucktensor/dasilva2013EAGEhtucktensor.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2013/dasilva2013EAGEhtucktensor/dasilva2013EAGEhtucktensor_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2013SINBADSPRhtuck,
	title = {Hierarchical Tucker tensor optimization - applications to 4D seismic data interpolation},
	year = {2013},
	publisher = {SINBAD},
	abstract = {In this work, we develop optimization algorithms on the manifold of Hierarchical Tucker (HT) tensors, an extremely efficient format for representing high-dimensional tensors exhibiting particular low-rank structure. With some minor alterations to existing theoretical developments, we develop an optimization framework based on the geometric understanding of HT tensors as a smooth manifold, a generalization of smooth curves/surfaces. Building on the existing research of solving optimization problems on smooth manifolds, we develop Steepest Descent and Conjugate Gradient methods for HT tensors. The resulting algorithms converge quickly, are immediately parallelizable, and do not require the computation of SVDs. We also derive efficient Gauss-Newton based algorithms which converge much faster than standard, first-order methods. We also extend ideas about favourable sampling conditions for missing-data recovery from the field of Matrix Completion to Tensor Completion and demonstrate how the organization of data can affect the success of recovery. As a result, if one has data with randomly missing source pairs, using these ideas, coupled with an efficient solver, one can interpolate large-scale seismic data volumes with missing sources and/or receivers by exploiting the multidimensional dependencies in the data. We are able to recover data volumes amidst extremely high subsampling ratios (in some cases, > 75\%) using this approach and we demonstrate our recovery on a synthetic 5D data set provided to us by BG.},
	keywords = {3D data interpolation, hierarchical tucker, Presentation, SINBAD, SINBADSPRING2013, SLIM, structured tensor},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Spring/dasilva2013SINBADSPRhtuck/dasilva2013SINBADSPRhtuck.pdf},
	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}
}
@presentation {dasilva2012SINBADhtt,
	title = {Hierarchical Tucker tensor optimization - applications to 4D seismic data interpolation},
	year = {2012},
	publisher = {SINBAD},
	abstract = {There has been a swell of research in the scientific computing community in the last couple of years which tries to extend notions of linear algebra (rank, the SVD, linear systems, etc.) to higher dimensional arrays, or tensors. Much work has been proposed to try to overcome the so called "curse of dimensionality", the O(N^d) storage required for a d-dimensional array, where N is the size of each dimension. The hierarchical Tucker format is one such tensor representation which manages to decompose a hierarchy of dimensions into parameter matrices of very manageable size, requiring at most dNK + (d - 2)K^3 + K^2 parameters, where K is an internal rank parameter. In this work, we extend ideas of matrix completion to the tensor case, where we only know a small number of randomly distributed entries from various 4D frequency slices, and try to recover the fully sampled tensor based on the knowledge that it has low hierarchical tucker rank in a particular arrangement of dimensions. Using this approach, we exploit the multi-dimensional dependencies within the full data in order to achieve very promising interpolation results even from heavily subsampled data.},
	keywords = {Presentation, SINBAD, SINBADFALL2012, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2012/Fall/dasilva2012SINBADhtt/dasilva2012SINBADhtt_pres.pdf},
	author = {Curt Da Silva}
}
