<s>
L1-norm	B-General_Concept
principal	I-General_Concept
component	I-General_Concept
analysis	I-General_Concept
(	O
L1-PCA	O
)	O
is	O
a	O
general	O
method	O
for	O
multivariate	O
data	O
analysis	O
.	O
</s>
<s>
L1-PCA	O
is	O
often	O
preferred	O
over	O
standard	O
L2-norm	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
when	O
the	O
analyzed	O
data	O
may	O
contain	O
outliers	O
(	O
faulty	O
values	O
or	O
corruptions	O
)	O
.	O
</s>
<s>
Both	O
L1-PCA	O
and	O
standard	O
PCA	O
seek	O
a	O
collection	O
of	O
orthogonal	B-Application
directions	O
(	O
principal	B-Application
components	I-Application
)	O
that	O
define	O
a	O
subspace	O
wherein	O
data	O
representation	O
is	O
maximized	O
according	O
to	O
the	O
selected	O
criterion	O
.	O
</s>
<s>
In	O
PCA	O
and	O
L1-PCA	O
,	O
the	O
number	O
of	O
principal	B-Application
components	I-Application
(	O
PCs	O
)	O
is	O
lower	O
than	O
the	O
rank	O
of	O
the	O
analyzed	O
matrix	B-Architecture
,	O
which	O
coincides	O
with	O
the	O
dimensionality	O
of	O
the	O
space	O
defined	O
by	O
the	O
original	O
data	O
points	O
.	O
</s>
<s>
Therefore	O
,	O
PCA	O
or	O
L1-PCA	O
are	O
commonly	O
employed	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
for	O
the	O
purpose	O
of	O
data	O
denoising	O
or	O
compression	O
.	O
</s>
<s>
Consider	O
any	O
matrix	B-Architecture
consisting	O
of	O
-dimensional	O
data	O
points	O
.	O
</s>
<s>
If	O
one	O
substitutes	O
in	O
(	O
)	O
by	O
the	O
Frobenius/L2	O
-norm	O
,	O
then	O
the	O
problem	O
becomes	O
standard	O
PCA	O
and	O
it	O
is	O
solved	O
by	O
the	O
matrix	B-Architecture
that	O
contains	O
the	O
dominant	O
singular	O
vectors	O
of	O
(	O
i.e.	O
,	O
the	O
singular	O
vectors	O
that	O
correspond	O
to	O
the	O
highest	O
singular	O
values	O
)	O
.	O
</s>
<s>
For	O
any	O
matrix	B-Architecture
with	O
,	O
define	O
as	O
the	O
nearest	O
(	O
in	O
the	O
L2-norm	O
sense	O
)	O
matrix	B-Architecture
to	O
that	O
has	O
orthonormal	O
columns	O
.	O
</s>
<s>
The	O
nuclear-norm	O
in	O
(	O
)	O
returns	O
the	O
summation	O
of	O
the	O
singular	O
values	O
of	O
its	O
matrix	B-Architecture
argument	O
and	O
can	O
be	O
calculated	O
by	O
means	O
of	O
standard	O
SVD	O
.	O
</s>
<s>
where	O
returns	O
the	O
-sign	O
matrix	B-Architecture
of	O
its	O
matrix	B-Architecture
argument	O
(	O
with	O
no	O
loss	O
of	O
generality	O
,	O
we	O
can	O
consider	O
)	O
.	O
</s>
<s>
L1-PCA	O
has	O
also	O
been	O
extended	O
for	O
the	O
analysis	O
of	O
tensor	B-Device
data	O
,	O
in	O
the	O
form	O
of	O
L1-Tucker	O
,	O
the	O
L1-norm	O
robust	O
analogous	O
of	O
standard	O
Tucker	B-Algorithm
decomposition	I-Algorithm
.	O
</s>
<s>
MATLAB	B-Language
code	O
for	O
L1-PCA	O
is	O
available	O
at	O
MathWorks	O
and	O
other	O
repositories	O
.	O
</s>
