<s>
Maximum	B-Algorithm
Variance	I-Algorithm
Unfolding	I-Algorithm
(	O
MVU	O
)	O
,	O
also	O
known	O
as	O
Semidefinite	B-Algorithm
Embedding	I-Algorithm
(	O
SDE	O
)	O
,	O
is	O
an	O
algorithm	O
in	O
computer	B-General_Concept
science	I-General_Concept
that	O
uses	O
semidefinite	O
programming	O
to	O
perform	O
non-linear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
of	O
high-dimensional	O
vectorial	O
input	O
data	O
.	O
</s>
<s>
It	O
is	O
motivated	O
by	O
the	O
observation	O
that	O
kernel	B-Algorithm
Principal	I-Algorithm
Component	I-Algorithm
Analysis	I-Algorithm
(	O
kPCA	O
)	O
does	O
not	O
reduce	O
the	O
data	O
dimensionality	O
,	O
as	O
it	O
leverages	O
the	O
Kernel	O
trick	O
to	O
non-linearly	O
map	O
the	O
original	O
data	O
into	O
an	O
inner-product	O
space	O
.	O
</s>
<s>
Let	O
be	O
the	O
Gram	B-Algorithm
matrices	I-Algorithm
of	O
and	O
(	O
i.e.	O
</s>
<s>
The	O
objective	O
function	O
can	O
be	O
rewritten	O
purely	O
in	O
the	O
form	O
of	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
After	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
learned	O
by	O
semidefinite	O
programming	O
,	O
the	O
output	O
can	O
be	O
obtained	O
via	O
Cholesky	O
decomposition	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
can	O
be	O
written	O
as	O
where	O
is	O
the	O
i-th	O
element	O
of	O
eigenvector	O
of	O
the	O
eigenvalue	O
.	O
</s>
