Hierarchical Tucker Tensor Optimization - Applications to Tensor Completion

Curt Da Silva just sent me the following:

Hey Igor,

..... I found your Nuit Blanche article on "A Riemannian geometry for low-rank matrix completion" [note from Igor: It's not mine it's Bamdev Mishra, K. Adithya Apuroop and Rodolphe Sepulchre's] and I would like to make you aware of an extension, in a similar vein to this work, to the multidimensional interpolation case in the Hierarchical Tucker format, which can be found at https://www.slim.eos.ubc.ca/content/hierarchical-tucker-tensor-optimization-applications-tensor-completion . We use a manifold-based approach to efficiently interpolate Hierarchical Tucker tensors with missing entries, specifically for seismic examples. We should be releasing a preprint of our full work in the near future as well. Thank you for your consideration.

Sincerely,

Curt Da Silva

Thanks Curt ! here is the paper: Hierarchical Tucker Tensor Optimization - Applications to Tensor Completion by Curt Da Silva, Felix J. Herrmann

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'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.

I wonder if the QTT format could be useful as well.



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