<s>
In	O
mathematics	O
,	O
Tucker	B-Algorithm
decomposition	I-Algorithm
decomposes	O
a	O
tensor	B-Device
into	O
a	O
set	O
of	O
matrices	O
and	O
one	O
small	O
core	O
tensor	B-Device
.	O
</s>
<s>
Initially	O
described	O
as	O
a	O
three-mode	O
extension	O
of	O
factor	O
analysis	O
and	O
principal	B-Application
component	I-Application
analysis	I-Application
it	O
may	O
actually	O
be	O
generalized	O
to	O
higher	O
mode	O
analysis	O
,	O
which	O
is	O
also	O
called	O
higher-order	O
singular	O
value	O
decomposition	O
(	O
HOSVD	O
)	O
.	O
</s>
<s>
In	O
PARAFAC	O
the	O
core	O
tensor	B-Device
is	O
restricted	O
to	O
be	O
"	O
diagonal	O
"	O
.	O
</s>
<s>
In	O
practice	O
,	O
Tucker	B-Algorithm
decomposition	I-Algorithm
is	O
used	O
as	O
a	O
modelling	O
tool	O
.	O
</s>
<s>
The	O
model	O
gives	O
a	O
summary	O
of	O
the	O
information	O
in	O
the	O
data	O
,	O
in	O
the	O
same	O
way	O
as	O
principal	B-Application
components	I-Application
analysis	I-Application
does	O
for	O
two-way	O
data	O
.	O
</s>
<s>
For	O
a	O
3rd-order	O
tensor	B-Device
,	O
where	O
is	O
either	O
or	O
,	O
Tucker	B-Algorithm
Decomposition	I-Algorithm
can	O
be	O
denoted	O
as	O
follows	O
,	O
</s>
<s>
where	O
is	O
the	O
core	O
tensor	B-Device
,	O
a	O
3rd-order	O
tensor	B-Device
that	O
contains	O
the	O
1-mode	O
,	O
2-mode	O
and	O
3-mode	O
singular	O
values	O
of	O
,	O
which	O
are	O
defined	O
as	O
the	O
Frobenius	O
norm	O
of	O
the	O
1-mode	O
,	O
2-mode	O
and	O
3-mode	O
slices	O
of	O
tensor	B-Device
respectively	O
.	O
</s>
<s>
There	O
are	O
two	O
special	O
cases	O
of	O
Tucker	B-Algorithm
decomposition	I-Algorithm
:	O
</s>
