Lecture 2

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Tensor formats ⊲ Hierarchical Tucker format (HT; Hackbusch/Kühn, Grasedyck, Kressner : Tree-tensor networks) ⊲ Tucker format (Q: MCTDH(F)) U(x1, .., x d) = {1,2,3,4,5} 1 2 3 4 5 r1� . . . k1=1 r d� k d =1 B(k1, .., k d) d� Ui(ki, xi) i=1
Tensor formats ⊲ Hierarchical Tucker format (HT; Hackbusch/Kühn, Grasedyck, Kressner : Tree-tensor networks) ⊲ Tucker format (Q: MCTDH(F)) ⊲ Tensor Train (TT-)format (Oseledets/Tyrtyshnikov, � MPS-format of quantum physics) U(x) = {1,2,3,4,5} r1� . . . k1=1 {1} {2,3,4,5} {2} {3,4,5} {3} {4} r d−1 � kd−1=1 i=1 {4,5} {5} d� Bi(ki−1, xi, ki) = B1(x1) · · · Bd(xd) U 1 U 2 U 3 U 4 U 5 r 1 r 2 r 3 r 4 n 1 n 2 n 3 n 4 n 5
Page 1 and 2: Tensor networks, TT (Matrix Product
Page 3 and 4: Contracted tensor representations E
Page 5 and 6: Contracted tensor representations D
Page 7 and 8: Tensor networks Diagramm: u [x] Dia
Page 9: Tensor formats ⊲ Hierarchical Tuc
Page 13 and 14: TT approximations of Friedman data
Page 15 and 16: Hierarchical Tucker as subspace app
Page 17 and 18: Orthonormal basis and recursive des
Page 19 and 20: Hierarchical Tucker tensors ⊲ Can
Page 25 and 26: HOSVD bases (Hackbusch) 1) Assume r
Page 27 and 28: Successive SVD for e.g. TT tensors
Page 29 and 30: Successive SVD for e.g. TT tensors
Page 31 and 32: Closedness A tree T D is characteri
Page 33 and 34: Summary For Tucker and HT redundanc

Lecture 2

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