An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples
Abstract: We review two similar concepts of hierarchical rank of tensors (which extend the matrix rank to higher order tensors): the TT-rank and the H-rank (hierarchical or H-Tucker rank). Based on this notion of rank, one can define a data-sparse representation of tensors involving O (dnk + dk^3) data for order d tensors with mode sizes n and rank k. Simple examples underline the differences and similarities between the different formats and ranks. Finally, we derive rank bounds for tensors in one of the formats based on the ranks in the other format.
- Location
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Deutsche Nationalbibliothek Frankfurt am Main
- Extent
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Online-Ressource
- Language
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Englisch
- Bibliographic citation
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An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples ; volume:11 ; number:3 ; year:2011 ; pages:291-304
Computational methods in applied mathematics ; 11, Heft 3 (2011), 291-304
- Creator
- DOI
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10.2478/cmam-2011-0016
- URN
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urn:nbn:de:101:1-2410261641042.771721468543
- Rights
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Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
- Last update
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15.08.2025, 7:28 AM CEST
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Associated
- Grasedyck, Lars
- Hackbusch, Wolfgang
Other Objects (12)
An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples
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Maximum relative distance between real rank-two and rank-one tensors
Maximum relative distance between real rank-two and rank-one tensors
On best rank one approximation of tensors
Rank and border rank of Kronecker powers of tensors and Strassen's laser method
Best rank-k approximations for tensors: generalizing Eckart–Young
FRAPPE: fast rank approximation with explainable features for tensors
Quantifying low rank approximations of third order symmetric tensors
An algorithm for the non-identifiability of rank-3 tensors
Correction: FRAPPE: fast rank approximation with explainable features for tensors
Some examples of rank-2 Brill–Noether loci
An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples
Identifiability of Rank-3 Tensors
Maximum relative distance between real rank-two and rank-one tensors
Maximum relative distance between real rank-two and rank-one tensors
On best rank one approximation of tensors
Rank and border rank of Kronecker powers of tensors and Strassen's laser method
Best rank-k approximations for tensors: generalizing Eckart–Young
FRAPPE: fast rank approximation with explainable features for tensors
Quantifying low rank approximations of third order symmetric tensors
An algorithm for the non-identifiability of rank-3 tensors
Correction: FRAPPE: fast rank approximation with explainable features for tensors
Some examples of rank-2 Brill–Noether loci
An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples
Identifiability of Rank-3 Tensors
Maximum relative distance between real rank-two and rank-one tensors
Maximum relative distance between real rank-two and rank-one tensors
On best rank one approximation of tensors
Rank and border rank of Kronecker powers of tensors and Strassen's laser method
Best rank-k approximations for tensors: generalizing Eckart–Young
FRAPPE: fast rank approximation with explainable features for tensors
Quantifying low rank approximations of third order symmetric tensors
An algorithm for the non-identifiability of rank-3 tensors
Correction: FRAPPE: fast rank approximation with explainable features for tensors