Lecture 2

Ranks of TT and HT tensors
The minimal numbers r i for a TT or HT respresentation of a
tensor U is well defined.
1. The optimal ranks r i, r = (r1, . . . , r d−1),
of the TT decomposition are equal to the ranks of the following
matrices, i = 1, . . . , d − 1
r i = rank of A i = U x i+1,...,x d
x1,...,x i
2. If ν ⊂ D then µ := D\µ the the mode ν-rank rν is given by
rν = rank of A i = U x µ
x ν .
(If ν = {1, 2, 7, 8} and µ = {3, 4, 5, 6} then x ν = (x1, x2, x7, x8). )
.