<s>
Sparse	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
sparse	B-Algorithm
PCA	I-Algorithm
)	O
is	O
a	O
specialised	O
technique	O
used	O
in	O
statistical	O
analysis	O
and	O
,	O
in	O
particular	O
,	O
in	O
the	O
analysis	O
of	O
multivariate	O
data	O
sets	O
.	O
</s>
<s>
It	O
extends	O
the	O
classic	O
method	O
of	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	B-Application
)	O
for	O
the	O
reduction	O
of	O
dimensionality	O
of	O
data	O
by	O
introducing	O
sparsity	O
structures	O
to	O
the	O
input	O
variables	O
.	O
</s>
<s>
A	O
particular	O
disadvantage	O
of	O
ordinary	O
PCA	B-Application
is	O
that	O
the	O
principal	B-Application
components	I-Application
are	O
usually	O
linear	O
combinations	O
of	O
all	O
input	O
variables	O
.	O
</s>
<s>
Sparse	B-Algorithm
PCA	I-Algorithm
overcomes	O
this	O
disadvantage	O
by	O
finding	O
linear	O
combinations	O
that	O
contain	O
just	O
a	O
few	O
input	O
variables	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
if	O
does	O
not	O
converge	O
to	O
zero	O
,	O
the	O
classical	O
PCA	B-Application
is	O
not	O
consistent	O
.	O
</s>
<s>
Consider	O
a	O
data	O
matrix	B-Architecture
,	O
,	O
where	O
each	O
of	O
the	O
columns	O
represent	O
an	O
input	O
variable	O
,	O
and	O
each	O
of	O
the	O
rows	O
represents	O
an	O
independent	O
sample	O
from	O
data	O
population	O
.	O
</s>
<s>
Let	O
be	O
the	O
empirical	O
covariance	O
matrix	B-Architecture
of	O
,	O
which	O
has	O
dimension	O
.	O
</s>
<s>
Given	O
an	O
integer	O
with	O
,	O
the	O
sparse	B-Algorithm
PCA	I-Algorithm
problem	O
can	O
be	O
formulated	O
as	O
maximizing	O
the	O
variance	O
along	O
a	O
direction	O
represented	O
by	O
vector	O
while	O
constraining	O
its	O
cardinality	O
:	O
</s>
<s>
If	O
one	O
takes	O
k	O
=p	O
,	O
the	O
problem	O
reduces	O
to	O
the	O
ordinary	O
PCA	B-Application
,	O
and	O
the	O
optimal	O
value	O
becomes	O
the	O
largest	O
eigenvalue	O
of	O
covariance	O
matrix	B-Architecture
Σ	O
.	O
</s>
<s>
and	O
iterate	O
this	O
process	O
to	O
obtain	O
further	O
principal	B-Application
components	I-Application
.	O
</s>
<s>
However	O
,	O
unlike	O
PCA	B-Application
,	O
sparse	B-Algorithm
PCA	I-Algorithm
cannot	O
guarantee	O
that	O
different	O
principal	B-Application
components	I-Application
are	O
orthogonal	O
.	O
</s>
<s>
The	O
following	O
equivalent	O
definition	O
is	O
in	O
matrix	B-Architecture
form	O
.	O
</s>
<s>
Tr	O
is	O
the	O
matrix	B-Architecture
trace	O
,	O
and	O
represents	O
the	O
non-zero	O
elements	O
in	O
matrix	B-Architecture
V	O
.	O
</s>
<s>
The	O
last	O
line	O
specifies	O
that	O
V	O
has	O
matrix	B-Architecture
rank	O
one	O
and	O
is	O
positive	B-Algorithm
semidefinite	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
sparse	B-Algorithm
PCA	I-Algorithm
problem	O
in	O
is	O
NP-hard	O
in	O
the	O
strong	O
sense	O
.	O
</s>
<s>
a	O
penelized	O
matrix	B-Architecture
decomposition	O
framework	O
,	O
</s>
<s>
The	O
methodological	O
and	O
theoretical	O
developments	O
of	O
Sparse	B-Algorithm
PCA	I-Algorithm
as	O
well	O
as	O
its	O
applications	O
in	O
scientific	O
studies	O
are	O
recently	O
reviewed	O
in	O
a	O
survey	O
paper	O
.	O
</s>
<s>
It	O
has	O
been	O
proposed	O
that	O
sparse	B-Algorithm
PCA	I-Algorithm
can	O
be	O
approximated	O
by	O
semidefinite	O
programming	O
(	O
SDP	O
)	O
.	O
</s>
<s>
In	O
the	O
second	O
constraint	O
,	O
is	O
a	O
p×1	O
vector	O
of	O
ones	O
,	O
and	O
|V|	O
is	O
the	O
matrix	B-Architecture
whose	O
elements	O
are	O
the	O
absolute	O
values	O
of	O
the	O
elements	O
of	O
V	O
.	O
</s>
<s>
Suppose	O
ordinary	O
PCA	B-Application
is	O
applied	O
to	O
a	O
dataset	O
where	O
each	O
input	O
variable	O
represents	O
a	O
different	O
asset	O
,	O
it	O
may	O
generate	O
principal	B-Application
components	I-Application
that	O
are	O
weighted	O
combination	O
of	O
all	O
the	O
assets	O
.	O
</s>
<s>
In	O
contrast	O
,	O
sparse	B-Algorithm
PCA	I-Algorithm
would	O
produce	O
principal	B-Application
components	I-Application
that	O
are	O
weighted	O
combination	O
of	O
only	O
a	O
few	O
input	O
assets	O
,	O
so	O
one	O
can	O
easily	O
interpret	O
its	O
meaning	O
.	O
</s>
<s>
Furthermore	O
,	O
if	O
one	O
uses	O
a	O
trading	O
strategy	O
based	O
on	O
these	O
principal	B-Application
components	I-Application
,	O
fewer	O
assets	O
imply	O
less	O
transaction	O
costs	O
.	O
</s>
<s>
Sparse	B-Algorithm
PCA	I-Algorithm
can	O
produce	O
a	O
principal	B-Application
component	I-Application
that	O
involves	O
only	O
a	O
few	O
genes	O
,	O
so	O
researchers	O
can	O
focus	O
on	O
these	O
specific	O
genes	O
for	O
further	O
analysis	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
if	O
does	O
not	O
converge	O
to	O
zero	O
,	O
the	O
classical	O
PCA	B-Application
is	O
not	O
consistent	O
.	O
</s>
<s>
Consider	O
a	O
hypothesis	O
test	O
where	O
the	O
null	O
hypothesis	O
specifies	O
that	O
data	O
are	O
generated	O
from	O
a	O
multivariate	O
normal	O
distribution	O
with	O
mean	O
0	O
and	O
covariance	O
equal	O
to	O
an	O
identity	O
matrix	B-Architecture
,	O
and	O
the	O
alternative	O
hypothesis	O
specifies	O
that	O
data	O
is	O
generated	O
from	O
a	O
spiked	O
model	O
with	O
signal	O
strength	O
:	O
</s>
<s>
scikit-learn	B-Application
–	O
Python	O
library	O
for	O
machine	O
learning	O
which	O
contains	O
Sparse	B-Algorithm
PCA	I-Algorithm
and	O
other	O
techniques	O
in	O
the	O
decomposition	O
module	O
.	O
</s>
