<s>
In	O
machine	O
learning	O
,	O
Manifold	B-General_Concept
regularization	I-General_Concept
is	O
a	O
technique	O
for	O
using	O
the	O
shape	O
of	O
a	O
dataset	O
to	O
constrain	O
the	O
functions	O
that	O
should	O
be	O
learned	O
on	O
that	O
dataset	O
.	O
</s>
<s>
The	O
technique	O
of	O
manifold	B-Architecture
learning	O
assumes	O
that	O
the	O
relevant	O
subset	O
of	O
data	O
comes	O
from	O
a	O
manifold	B-Architecture
,	O
a	O
mathematical	O
structure	O
with	O
useful	O
properties	O
.	O
</s>
<s>
Because	O
of	O
this	O
assumption	O
,	O
a	O
manifold	B-General_Concept
regularization	I-General_Concept
algorithm	O
can	O
use	O
unlabeled	O
data	O
to	O
inform	O
where	O
the	O
learned	O
function	O
is	O
allowed	O
to	O
change	O
quickly	O
and	O
where	O
it	O
is	O
not	O
,	O
using	O
an	O
extension	O
of	O
the	O
technique	O
of	O
Tikhonov	O
regularization	O
.	O
</s>
<s>
Manifold	B-General_Concept
regularization	I-General_Concept
algorithms	O
can	O
extend	O
supervised	B-General_Concept
learning	I-General_Concept
algorithms	O
in	O
semi-supervised	B-General_Concept
learning	I-General_Concept
and	O
transductive	B-General_Concept
learning	I-General_Concept
settings	O
,	O
where	O
unlabeled	O
data	O
are	O
available	O
.	O
</s>
<s>
The	O
technique	O
has	O
been	O
used	O
for	O
applications	O
including	O
medical	B-Application
imaging	I-Application
,	O
geographical	O
imaging	O
,	O
and	O
object	O
recognition	O
.	O
</s>
<s>
Manifold	B-General_Concept
regularization	I-General_Concept
is	O
a	O
type	O
of	O
regularization	O
,	O
a	O
family	O
of	O
techniques	O
that	O
reduces	O
overfitting	B-Error_Name
and	O
ensures	O
that	O
a	O
problem	O
is	O
well-posed	B-Algorithm
by	O
penalizing	O
complex	O
solutions	O
.	O
</s>
<s>
In	O
particular	O
,	O
manifold	B-General_Concept
regularization	I-General_Concept
extends	O
the	O
technique	O
of	O
Tikhonov	O
regularization	O
as	O
applied	O
to	O
Reproducing	O
kernel	B-Algorithm
Hilbert	O
spaces	O
(	O
RKHSs	O
)	O
.	O
</s>
<s>
The	O
hypothesis	O
space	O
is	O
an	O
RKHS	O
,	O
meaning	O
that	O
it	O
is	O
associated	O
with	O
a	O
kernel	B-Algorithm
,	O
and	O
so	O
every	O
candidate	O
function	O
has	O
a	O
norm	O
,	O
which	O
represents	O
the	O
complexity	O
of	O
the	O
candidate	O
function	O
in	O
the	O
hypothesis	O
space	O
.	O
</s>
<s>
where	O
is	O
a	O
hyperparameter	B-General_Concept
that	O
controls	O
how	O
much	O
the	O
algorithm	O
will	O
prefer	O
simpler	O
functions	O
over	O
functions	O
that	O
fit	O
the	O
data	O
better	O
.	O
</s>
<s>
Manifold	B-General_Concept
regularization	I-General_Concept
adds	O
a	O
second	O
regularization	O
term	O
,	O
the	O
intrinsic	O
regularizer	O
,	O
to	O
the	O
ambient	O
regularizer	O
used	O
in	O
standard	O
Tikhonov	O
regularization	O
.	O
</s>
<s>
Under	O
the	O
manifold	B-General_Concept
assumption	I-General_Concept
in	O
machine	O
learning	O
,	O
the	O
data	O
in	O
question	O
do	O
not	O
come	O
from	O
the	O
entire	O
input	O
space	O
,	O
but	O
instead	O
from	O
a	O
nonlinear	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
geometry	O
of	O
this	O
manifold	B-Architecture
,	O
the	O
intrinsic	O
space	O
,	O
is	O
used	O
to	O
determine	O
the	O
regularization	O
norm	O
.	O
</s>
<s>
Many	O
natural	O
choices	O
involve	O
the	O
gradient	B-Language
on	I-Language
the	I-Language
manifold	I-Language
,	O
which	O
can	O
provide	O
a	O
measure	O
of	O
how	O
smooth	O
a	O
target	O
function	O
is	O
.	O
</s>
<s>
When	O
the	O
distances	O
between	O
input	O
points	O
are	O
interpreted	O
as	O
a	O
graph	O
,	O
then	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
of	O
the	O
graph	O
can	O
help	O
to	O
estimate	O
the	O
marginal	O
distribution	O
.	O
</s>
<s>
Define	O
to	O
be	O
a	O
diagonal	O
matrix	O
with	O
and	O
to	O
be	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Then	O
,	O
as	O
the	O
number	O
of	O
data	O
points	O
increases	O
,	O
converges	O
to	O
the	O
Laplace	O
–	O
Beltrami	O
operator	O
,	O
which	O
is	O
the	O
divergence	B-Application
of	O
the	O
gradient	O
.	O
</s>
<s>
As	O
with	O
other	O
kernel	B-Algorithm
methods	I-Algorithm
,	O
may	O
be	O
an	O
infinite-dimensional	O
space	O
,	O
so	O
if	O
the	O
regularization	O
expression	O
cannot	O
be	O
solved	O
explicitly	O
,	O
it	O
is	O
impossible	O
to	O
search	O
the	O
entire	O
space	O
for	O
a	O
solution	O
.	O
</s>
<s>
Instead	O
,	O
a	O
representer	O
theorem	O
shows	O
that	O
under	O
certain	O
conditions	O
on	O
the	O
choice	O
of	O
the	O
norm	O
,	O
the	O
optimal	O
solution	O
must	O
be	O
a	O
linear	O
combination	O
of	O
the	O
kernel	B-Algorithm
centered	O
at	O
each	O
of	O
the	O
input	O
points	O
:	O
for	O
some	O
weights	O
,	O
</s>
<s>
This	O
method	O
is	O
akin	O
local	B-General_Concept
averaging	I-General_Concept
methods	I-General_Concept
,	O
that	O
are	O
known	O
to	O
scale	O
poorly	O
in	O
high-dimensional	O
problem	O
.	O
</s>
<s>
Indeed	O
,	O
graph	B-Algorithm
Laplacian	I-Algorithm
is	O
known	O
to	O
suffer	O
from	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
This	O
method	O
consists	O
in	O
estimating	O
the	O
Laplacian	O
operator	O
thanks	O
to	O
derivatives	O
of	O
the	O
kernel	B-Algorithm
reading	O
where	O
denotes	O
the	O
partial	O
derivatives	O
according	O
to	O
the	O
j-th	O
coordinate	O
of	O
the	O
first	O
variable	O
.	O
</s>
<s>
This	O
second	O
approach	O
of	O
the	O
Laplacian	O
norm	O
is	O
to	O
put	O
in	O
relation	O
with	O
meshfree	B-Algorithm
methods	I-Algorithm
,	O
that	O
contrast	O
with	O
the	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
in	O
PDE	O
.	O
</s>
<s>
Manifold	B-General_Concept
regularization	I-General_Concept
can	O
extend	O
a	O
variety	O
of	O
algorithms	O
that	O
can	O
be	O
expressed	O
using	O
Tikhonov	O
regularization	O
,	O
by	O
choosing	O
an	O
appropriate	O
loss	O
function	O
and	O
hypothesis	O
space	O
.	O
</s>
<s>
Two	O
commonly	O
used	O
examples	O
are	O
the	O
families	O
of	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
and	O
regularized	O
least	O
squares	O
algorithms	O
.	O
</s>
<s>
(	O
Regularized	O
least	O
squares	O
includes	O
the	O
ridge	O
regression	O
algorithm	O
;	O
the	O
related	O
algorithms	O
of	O
LASSO	O
and	O
elastic	O
net	O
regularization	O
can	O
be	O
expressed	O
as	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
.	O
)	O
</s>
<s>
The	O
extended	O
versions	O
of	O
these	O
algorithms	O
are	O
called	O
Laplacian	O
Regularized	O
Least	O
Squares	O
(	O
abbreviated	O
LapRLS	O
)	O
and	O
Laplacian	O
Support	B-Algorithm
Vector	I-Algorithm
Machines	I-Algorithm
(	O
LapSVM	O
)	O
,	O
respectively	O
.	O
</s>
<s>
In	O
particular	O
,	O
RLS	O
is	O
designed	O
to	O
minimize	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
between	O
the	O
predicted	O
values	O
and	O
the	O
true	O
labels	O
,	O
subject	O
to	O
regularization	O
.	O
</s>
<s>
Ridge	O
regression	O
is	O
one	O
form	O
of	O
RLS	O
;	O
in	O
general	O
,	O
RLS	O
is	O
the	O
same	O
as	O
ridge	O
regression	O
combined	O
with	O
the	O
kernel	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
statement	O
for	O
RLS	O
results	O
from	O
choosing	O
the	O
loss	O
function	O
in	O
Tikhonov	O
regularization	O
to	O
be	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
:	O
</s>
<s>
Thanks	O
to	O
the	O
representer	O
theorem	O
,	O
the	O
solution	O
can	O
be	O
written	O
as	O
a	O
weighted	O
sum	O
of	O
the	O
kernel	B-Algorithm
evaluated	O
at	O
the	O
data	O
points	O
:	O
</s>
<s>
where	O
is	O
defined	O
to	O
be	O
the	O
kernel	B-Algorithm
matrix	O
,	O
with	O
,	O
and	O
is	O
the	O
vector	O
of	O
data	O
labels	O
.	O
</s>
<s>
Adding	O
a	O
Laplacian	O
term	O
for	O
manifold	B-General_Concept
regularization	I-General_Concept
gives	O
the	O
Laplacian	O
RLS	O
statement	O
:	O
</s>
<s>
Letting	O
be	O
the	O
kernel	B-Algorithm
matrix	O
as	O
above	O
,	O
be	O
the	O
vector	O
of	O
data	O
labels	O
,	O
and	O
be	O
the	O
block	O
matrix	O
:	O
</s>
<s>
medical	B-Application
imaging	I-Application
,	O
</s>
<s>
document	B-Algorithm
classification	I-Algorithm
,	O
</s>
<s>
Support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
(	O
SVMs	B-Algorithm
)	O
are	O
a	O
family	O
of	O
algorithms	O
often	O
used	O
for	O
classifying	B-General_Concept
data	I-General_Concept
into	O
two	O
or	O
more	O
groups	O
,	O
or	O
classes	O
.	O
</s>
<s>
Intuitively	O
,	O
an	O
SVM	B-Algorithm
draws	O
a	O
boundary	O
between	O
classes	O
so	O
that	O
the	O
closest	O
labeled	O
examples	O
to	O
the	O
boundary	O
are	O
as	O
far	O
away	O
as	O
possible	O
.	O
</s>
<s>
This	O
can	O
be	O
directly	O
expressed	O
as	O
a	O
linear	B-Algorithm
program	I-Algorithm
,	O
but	O
it	O
is	O
also	O
equivalent	O
to	O
Tikhonov	O
regularization	O
with	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
function	O
,	O
:	O
</s>
<s>
Again	O
,	O
the	O
representer	O
theorem	O
allows	O
the	O
solution	O
to	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
kernel	B-Algorithm
evaluated	O
at	O
the	O
data	O
points	O
:	O
</s>
<s>
can	O
be	O
found	O
by	O
writing	O
the	O
problem	O
as	O
a	O
linear	B-Algorithm
program	I-Algorithm
and	O
solving	O
the	O
dual	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
medical	B-Application
imaging	I-Application
,	O
</s>
<s>
and	O
brain	B-Application
–	I-Application
computer	I-Application
interfaces	I-Application
.	O
</s>
<s>
Manifold	B-General_Concept
regularization	I-General_Concept
assumes	O
that	O
data	O
with	O
different	O
labels	O
are	O
not	O
likely	O
to	O
be	O
close	O
together	O
.	O
</s>
<s>
Depending	O
on	O
the	O
structure	O
of	O
the	O
data	O
,	O
it	O
may	O
be	O
necessary	O
to	O
use	O
a	O
different	O
semi-supervised	O
or	O
transductive	B-General_Concept
learning	I-General_Concept
algorithm	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
unlabeled	O
data	O
have	O
no	O
effect	O
on	O
the	O
solution	O
learned	O
by	O
manifold	B-General_Concept
regularization	I-General_Concept
,	O
even	O
if	O
the	O
data	O
fit	O
the	O
algorithm	O
's	O
assumption	O
that	O
the	O
separator	O
should	O
be	O
smooth	O
.	O
</s>
<s>
Approaches	O
related	O
to	O
co-training	B-Algorithm
have	O
been	O
proposed	O
to	O
address	O
this	O
limitation	O
.	O
</s>
<s>
If	O
there	O
are	O
a	O
very	O
large	O
number	O
of	O
unlabeled	O
examples	O
,	O
the	O
kernel	B-Algorithm
matrix	O
becomes	O
very	O
large	O
,	O
and	O
a	O
manifold	B-General_Concept
regularization	I-General_Concept
algorithm	O
may	O
become	O
prohibitively	O
slow	O
to	O
compute	O
.	O
</s>
<s>
Online	O
algorithms	O
and	O
sparse	O
approximations	O
of	O
the	O
manifold	B-Architecture
may	O
help	O
in	O
this	O
case	O
.	O
</s>
