Dynamical Tensor Approximation

By Othmar Koch, Christian Lubich. 2010

136 citations recorded by the source corpus.

Subfields: Computational Mathematics, Radiology, Nuclear Medicine and Imaging

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In Computational Mathematics

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Initially proposed for low-rank matrix manifolds in [166], dynamical low-rank methods have been extended to low-rank tensors in Tucker [167, 199], TT/MPS [118, 123, 163, 186], and HT [9, 186, 255] decomposition.”
A projector-splitting integrator for dynamical low-rank approximation (2013, unknown): “The approach and its error analysis were extended to tensors in the Tucker format in [7].”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “The original approach of Koch and Lubich [2007] (later generalized to the Tucker and TT models [Koch and Lubich, 2010]) is to write down ordinary differential equations for the parameters U(t), S(t), V(t) of the SVD like decomposition in the form Y(t) = U(t)S(t)VT(t).”
Efficient Low-rank Multimodal Fusion With Modality-Specific Factors (2018, methodology): “Razenshteyn et al. (2016) proposes a mod-iﬁed weighted version of low-rank approximation, and Koch and Lubich (2010) applies the method towards temporally dependent data to obtain low-rank approximations.”
Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors (2013, methodology): “The proofs are the same as for the corresponding results in [15] and [16], using Lemma 4.”
The geometry of algorithms using hierarchical tensors (2013, methodology): “A similar tensor was used in the context of Tucker tensors in [38].”
Zur Theorie der Niedrigrangapproximation in Tensorprodukten von Hilberträumen (2013, background): “Lubich [49], obwohl wir nirgends einen „strengen“ Beweis gefunden haben.”
Algorithm 941 (2014, background): “The second strategy is to reformulate the problem at hand as an optimization problem with the admissible set of solutions restricted to data-sparse tensors, see [8, 16, 17, 22, 29, 30] and the references therein.”
On the approximation of high-dimensional differential equations in the hierarchical Tucker format (2013, methodology): “Similar assumptions have been made in [8] where the approximation error of the non-hierarchical Tucker format has been analyzed.”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, background): “The second strategy is to reformulate the problem at hand as an optimization problem with the admissible set of solutions restricted to data-sparse tensors, see [21, 35, 36, 45, 63, 68] and the references therein.”
ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, unknown): “, [35]): we solve the problem minx̄:k(x̄)=k ‖dτ(x̄)/dt− dy/dt‖ , projecting the exact velocity dy/dt onto the tangent space of the TT manifold.”
Low-Rank Tensor Learning with Discriminant Analysis for Action Classification and Image Recovery (2014, background): “Many tensor representation learning approaches have been proposed (Chen and Saad 2009; Koch and Lubich 2010; Haegeman et al. 2011; Holtz, Rohwedder, and Schneider 2012; Khoromskij, Oseledets, and Schneider 2012; Arnold and Jahnke 2012; Lubich et al. 2013; Uschmajew and Vandereycken 2013; Mu et al.…”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “, [35]): we solve the problem minx̄:k(x̄)=k ‖dτ(x̄)/dt− dy/dt‖ , projecting the exact velocity dy/dt onto the tangent space of the TT manifold.”
Low-rank tensor completion by Riemannian optimization (2014, methodology): “The approach proposed in this paper is based on the observation that the set of tensors of fixed multilinear rank r, denoted by Mr, forms a smooth manifold [14, 28].”