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a
b
{5}
{5}
{4,5}
{4,5}
{3,4,5}
{4}
{3,4,5}
{4}
{2,…,5}
{3}
{1,…,5}
{3}
{1,…,5}
{2}
{1,2}
{1}
Fig. 9.6 Two examples of hierarchical decompositions of the index set for d
¼
5
suppose that we subdivide our index set into two subsets where the first
k
form the first
subset and the others the second. Then we arrive at the canonic dyadic decomposition
of the core tensor
c
i
1
,
...
,
i
k
;i
kþ
1
,
...
,
i
d
¼
X
t
c
i
1
,
...
,
i
k
,
s
c
i
kþ
1
,
...
,
i
d
,
s
:
ð
9
:
11
Þ
s¼
1
This reduces the
d-
dimensional core tensor to the new tensors of dimensions
k+1
and
d-k + 1
. The resulting Tucker tensor is
a
i
1
,
...
,
i
d
¼
X
j
1
¼
1
X
X
t
t
t
c
i
1
,
...
,
i
k
,
s
c
i
kþ
1
,
...
,
i
d
,
s
u
i
1
,
j
1
u
i
d
,
j
d
j
d
¼
1
s¼
1
By successively applying decompositions (
9.11
), we arrive at a hierarchical
Tucker decomposition. If we construct the tree such that the index sets of all leafs
contain one index only (Fig.
9.6
), for the resulting Tucker decomposition, the
dependence on
d
is linear!
Now it is possible to prove that for hierarchical decompositions like the
H-SVD, important properties of the Tucker decomposition are retained
[Gra10]. This includes the availability of standard computation algorithms, effi-
cient truncation, and stability. We will study this in more detail in the next section
which is devoted to an important type of hierarchical Tucker decompositions - the
tensor train.
9.3.2 Tensor-Train Decomposition
A hierarchical decomposition of the index set defined by a binary three where each
split is done into a node corresponding to a single index and into a node of remaining
