f¨ur Mathematik in den Naturwissenschaften Leipzig Hierarchical Singular Value Decomposition of Tensors

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “An alternative way to reduce the complexity of the Tucker decomposition is given by the hierarchical Tucker (HT) decomposition [103,117] (also called hierarchical tensor representation ).”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, background): “…[8] [9] [10] [11], Tucker [12] [13] [14] [15], Matrix Product States [16] [17] [18] [19] and MCTDH [20] in quantum physics community, HT (Hierarchical Tucker) [21] [22] and TT (Tensor Train) in numerical algebra community [23] [24] [25], and later the QTT (Quantized TT) [26] [27] [28] [29] [30].”
Algorithm 941 (2014, background): “The resulting tensor X̃ satisfies the following error bound [Grasedyck 2010b, Theorem 3.11] ∥∥X − X̃∥∥2 ≤ √∑ t∈T ′ δrt(X(t))2 ≤ √ 2d − 3 ∥∥X − Xbest∥∥2, (17) where Xbest represents the best approximation of X in H-Tucker((rt)t∈T ), T ′ := T \{troot, tchild} where tchild is a child of the root node…”
Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems (2011, methodology): “In the following, we provide a very brief description of the hierarchical Tucker decomposition (HTD) and refer to [8, 20] for more details.”
Fast evaluation of singular BEM integrals based on tensor approximations (2012, methodology): “The author thanks Prof. Dr. Dr. h.c. Wolfgang Hackbusch (Max Planck Institute for Mathematics in the Sciences, Leipzig) and Prof. Dr. Lars Grasedyck (RWTH Aachen) for helpful discussions.”
Black box approximation of tensors in hierarchical Tucker format (2013, methodology): “, nd ∈ N, in a data-sparse low rank tensor format, namely the hierarchical (H-) Tucker format [6, 5].”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, methodology): “MSc ETH CSE born July 31, 1984 citizen of Thal SG accepted on the recommendation of Prof. Dr. Daniel Kressner, examiner Prof. Dr. Lars Grasedyck, co-examiner Prof. Dr. Christoph Schwab, co-examiner 2012”
Tensor-Train Decomposition (2011, methodology): “3) is just another form of writing the tree format of [32] or the subspace approach of [18, 15]: the same threedimensional tensors as defining parameters, the same complexity estimates.”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, methodology): “Note that the straightforward discretizations of the Laplace operator we consider turn out to be very efficient in tensor-structured solvers based on the TT, QTT, and hierarchical tensor [15, 13] formats and yield convincing experimental results when constructed on fine meshes; see the papers [24, 8”
Tensor Ring Decomposition (2016, background): “Another important tensor network is the hierarchical Tucker (HT) format [33], [34], in which a recursive, hierarchical construction of Tucker type is employed for low-dimensional representation.”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig a Projection Method to Solve Linear Systems in Tensor Format a Projection Method to Solve Linear Systems in Tensor Format (unknown): “For the definition of the hierarchical Tucker format, we adopt the notation from [9].”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, background): “In a parallel effort, mathematicians have also investigated different concepts to apply tensor trains (which are analogous to MPS) to diverse problems (Grasedyck, 2010; Hackbusch, 2012; Oseledets, 2011).”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, unknown): “1 Hierarchical Tucker (HT) and Tensor Tree Network State (TTNS) Models Hierarchical Tucker (HT) decompositions (also called hierarchical tensor representation) have been introduced in [113] and also independently in [106], see also [108, 158, 181, 263] and references therein8.”
Tensors for Data Mining and Data Fusion (2016, methodology): “The ﬁrst way is the so-called Hierarchical Tucker Decomposition (H-Tucker) [Hackbusch and K¨uhn 2009; Grasedyck 2010; Kressner and Tobler 2012; Ballani et al. 2013].”
LOCAL CONVERGENCE OF ALTERNATING SCHEMES FOR OPTIMIZATION OF CONVEX PROBLEMS IN THE TT FORMAT (2011, methodology): “From the practical side, two notable special cases of the hierarchichal ansatz have received major attention: The HT format as used by Grasedyck [14], using a symmetric decomposition tree, and the TT format, as introduced independently by Tyrtyshnikov and Oseledets [30, 32], using a linear decomposi”
Efficient Analysis of High Dimensional Data in Tensor Formats (2012, methodology): “(49) This recursion is motivated through application of the Newton method on the function f(x) := u − x, see [20].”
Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (2012, methodology): “An analogue of hosvd for the H–Tucker format, which we shall henceforth abbreviate as H–hosvd, has been introduced in [64].”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, methodology): “The rst attempt used the hierarchical splitting of dimensions (see the hierarchical-type dimension splitting in [20], the H-Tucker format [12, 10] and the Tree-Tucker format [43, 38]), which are free from the curse of dimensionality, and what is important, basic MLA operations can be implemented in ”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “4) In practice, for very large scale problems, tensors are expressed approximately in tensor network formats, especially using Canonical Polyadic (CP), Tucker or Tensor Train (TT)/Hierarchical Tucker (HT) models (Oseledets, 2011a; Grasedyck, 2010).”