Lecture 2

Noteable special case of HT:
TT format (Oseledets & Tyrtyshnikov, 2009)
(matrix product states (MPS), Vidal 2003, Schöllwock et al.)
TT tensor U can be written as matrix product form
=
r1�
..
k1=1
rd−1 �
kd−1=1
U(x) = U1(x1) · · · U i(x i) · · · U d(x d)
U1(x1, k1)U2(k1, x1, k2)...Ud−1(kd−2xd−1, kd−1)Ud(kd−1, xd, kd)
with matrices U i(x i) = � u k i
k i−1 (x i) � ∈ R r i−1×r i , r0 = r d := 1
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
Redundancy: U(x) = U1(x1)GG −1 U2(x2) · · · U i(x i) · · · U d(x d) .