<s>
In	O
machine	O
learning	O
,	O
the	O
kernel	B-General_Concept
embedding	I-General_Concept
of	I-General_Concept
distributions	I-General_Concept
(	O
also	O
called	O
the	O
kernel	O
mean	O
or	O
mean	O
map	O
)	O
comprises	O
a	O
class	O
of	O
nonparametric	B-General_Concept
methods	O
in	O
which	O
a	O
probability	O
distribution	O
is	O
represented	O
as	O
an	O
element	O
of	O
a	O
reproducing	O
kernel	O
Hilbert	O
space	O
(	O
RKHS	O
)	O
.	O
</s>
<s>
A	O
generalization	O
of	O
the	O
individual	O
data-point	O
feature	O
mapping	O
done	O
in	O
classical	O
kernel	B-Algorithm
methods	I-Algorithm
,	O
the	O
embedding	O
of	O
distributions	O
into	O
infinite-dimensional	O
feature	O
spaces	O
can	O
preserve	O
all	O
of	O
the	O
statistical	O
features	O
of	O
arbitrary	O
distributions	O
,	O
while	O
allowing	O
one	O
to	O
compare	O
and	O
manipulate	O
distributions	O
using	O
Hilbert	O
space	O
operations	O
such	O
as	O
inner	O
products	O
,	O
distances	O
,	O
projections	B-Algorithm
,	O
linear	B-Architecture
transformations	I-Architecture
,	O
and	O
spectral	O
analysis	O
.	O
</s>
<s>
For	O
example	O
,	O
various	O
kernels	O
have	O
been	O
proposed	O
for	O
learning	O
from	O
data	O
which	O
are	O
:	O
vectors	O
in	O
,	O
discrete	O
classes/categories	O
,	O
strings	O
,	O
graphs/networks	O
,	O
images	O
,	O
time	O
series	O
,	O
manifolds	B-Architecture
,	O
dynamical	O
systems	O
,	O
and	O
other	O
structured	O
objects	O
.	O
</s>
<s>
A	O
review	O
of	O
recent	O
works	O
on	O
kernel	B-General_Concept
embedding	I-General_Concept
of	I-General_Concept
distributions	I-General_Concept
can	O
be	O
found	O
in	O
.	O
</s>
<s>
The	O
analysis	O
of	O
distributions	O
is	O
fundamental	O
in	O
machine	O
learning	O
and	O
statistics	O
,	O
and	O
many	O
algorithms	O
in	O
these	O
fields	O
rely	O
on	O
information	O
theoretic	O
approaches	O
such	O
as	O
entropy	B-Algorithm
,	O
mutual	O
information	O
,	O
or	O
Kullback	O
–	O
Leibler	O
divergence	O
.	O
</s>
<s>
Gaussian	O
mixture	O
models	O
)	O
,	O
while	O
nonparametric	B-General_Concept
methods	O
like	O
kernel	B-General_Concept
density	I-General_Concept
estimation	I-General_Concept
(	O
Note	O
:	O
the	O
smoothing	O
kernels	O
in	O
this	O
context	O
have	O
a	O
different	O
interpretation	O
than	O
the	O
kernels	O
discussed	O
here	O
)	O
or	O
characteristic	O
function	O
representation	O
(	O
via	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
distribution	O
)	O
break	O
down	O
in	O
high-dimensional	O
settings	O
.	O
</s>
<s>
Methods	O
based	O
on	O
the	O
kernel	B-General_Concept
embedding	I-General_Concept
of	I-General_Concept
distributions	I-General_Concept
sidestep	O
these	O
problems	O
and	O
also	O
possess	O
the	O
following	O
advantages	O
:	O
</s>
<s>
Thus	O
,	O
learning	O
via	O
the	O
kernel	B-General_Concept
embedding	I-General_Concept
of	I-General_Concept
distributions	I-General_Concept
offers	O
a	O
principled	O
drop-in	O
replacement	O
for	O
information	O
theoretic	O
approaches	O
and	O
is	O
a	O
framework	O
which	O
not	O
only	O
subsumes	O
many	O
popular	O
methods	O
in	O
machine	O
learning	O
and	O
statistics	O
as	O
special	O
cases	O
,	O
but	O
also	O
can	O
lead	O
to	O
entirely	O
new	O
learning	O
algorithms	O
.	O
</s>
<s>
Nevertheless	O
,	O
even	O
in	O
cases	O
where	O
the	O
assumption	O
fails	O
,	O
may	O
still	O
be	O
used	O
to	O
approximate	O
the	O
conditional	O
kernel	O
embedding	O
and	O
in	O
practice	O
,	O
the	O
inversion	O
operator	O
is	O
replaced	O
with	O
a	O
regularized	O
version	O
of	O
itself	O
(	O
where	O
denotes	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
)	O
.	O
</s>
<s>
where	O
are	O
implicitly	O
formed	O
feature	O
matrices	O
,	O
is	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
for	O
samples	O
of	O
,	O
and	O
is	O
a	O
regularization	O
parameter	O
needed	O
to	O
avoid	O
overfitting	B-Error_Name
.	O
</s>
<s>
In	O
the	O
presence	O
of	O
large	O
sample	O
sizes	O
,	O
manipulations	O
of	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
may	O
be	O
computationally	O
demanding	O
.	O
</s>
<s>
Through	O
use	O
of	O
a	O
low-rank	O
approximation	O
of	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
(	O
such	O
as	O
the	O
incomplete	O
Cholesky	O
factorization	O
)	O
,	O
running	O
time	O
and	O
memory	O
requirements	O
of	O
kernel-embedding-based	O
learning	O
algorithms	O
can	O
be	O
drastically	O
reduced	O
without	O
suffering	O
much	O
loss	O
in	O
approximation	O
accuracy	O
.	O
</s>
<s>
If	O
is	O
defined	O
such	O
that	O
takes	O
values	O
in	O
for	O
all	O
with	O
(	O
as	O
is	O
the	O
case	O
for	O
the	O
widely	O
used	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
kernels	O
)	O
,	O
then	O
with	O
probability	O
at	O
least	O
:	O
</s>
<s>
Statistics	O
based	O
on	O
kernel	O
embeddings	O
thus	O
avoid	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
,	O
and	O
though	O
the	O
true	O
underlying	O
distribution	O
is	O
unknown	O
in	O
practice	O
,	O
one	O
can	O
(	O
with	O
high	O
probability	O
)	O
obtain	O
an	O
approximation	O
within	O
of	O
the	O
true	O
kernel	O
embedding	O
based	O
on	O
a	O
finite	O
sample	O
of	O
size	O
.	O
</s>
<s>
If	O
induces	O
a	O
strictly	O
positive	B-Algorithm
definite	I-Algorithm
kernel	O
matrix	O
for	O
any	O
set	O
of	O
distinct	O
points	O
,	O
then	O
it	O
is	O
a	O
universal	O
kernel	O
.	O
</s>
<s>
For	O
example	O
,	O
Gaussian	O
RBF	O
is	O
universal	O
,	O
sinc	B-Algorithm
kernel	O
is	O
not	O
universal	O
.	O
</s>
<s>
One	O
can	O
thus	O
select	O
the	O
regularization	O
parameter	O
by	O
performing	O
cross-validation	B-Application
based	O
on	O
the	O
squared	O
loss	O
function	O
of	O
the	O
regression	O
problem	O
.	O
</s>
<s>
is	O
the	O
empirical	O
kernel	O
embedding	O
of	O
the	O
prior	O
distribution	O
,	O
,	O
and	O
are	O
Gram	B-Algorithm
matrices	O
with	O
entries	O
respectively	O
.	O
</s>
<s>
The	O
latter	O
regularization	O
is	O
done	O
on	O
square	O
of	O
because	O
may	O
not	O
be	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
where	O
the	O
maximization	O
is	O
done	O
over	O
the	O
entire	O
space	O
of	O
distributions	O
on	O
Here	O
,	O
is	O
the	O
kernel	O
embedding	O
of	O
the	O
proposed	O
density	O
and	O
is	O
an	O
entropy-like	O
quantity	O
(	O
e.g.	O
</s>
<s>
Entropy	B-Algorithm
,	O
KL	O
divergence	O
,	O
Bregman	B-Algorithm
divergence	I-Algorithm
)	O
.	O
</s>
<s>
Connections	O
between	O
the	O
ideas	O
underlying	O
Gaussian	B-General_Concept
processes	I-General_Concept
and	O
conditional	B-General_Concept
random	I-General_Concept
fields	I-General_Concept
may	O
be	O
drawn	O
with	O
the	O
estimation	O
of	O
conditional	O
probability	O
distributions	O
in	O
this	O
fashion	O
,	O
if	O
one	O
views	O
the	O
feature	O
mappings	O
associated	O
with	O
the	O
kernel	O
as	O
sufficient	O
statistics	O
in	O
generalized	O
(	O
possibly	O
infinite-dimensional	O
)	O
exponential	O
families	O
.	O
</s>
<s>
samples	O
of	O
each	O
random	O
variable	O
,	O
a	O
simple	O
parameter-free	O
unbiased	O
estimator	O
of	O
HSIC	O
which	O
exhibits	O
concentration	O
about	O
the	O
true	O
value	O
can	O
be	O
computed	O
in	O
time	O
,	O
where	O
the	O
Gram	B-Algorithm
matrices	O
of	O
the	O
two	O
datasets	O
are	O
approximated	O
using	O
with	O
.	O
</s>
<s>
The	O
desirable	O
properties	O
of	O
HSIC	O
have	O
led	O
to	O
the	O
formulation	O
of	O
numerous	O
algorithms	O
which	O
utilize	O
this	O
dependence	O
measure	O
for	O
a	O
variety	O
of	O
common	O
machine	O
learning	O
tasks	O
such	O
as	O
:	O
feature	B-General_Concept
selection	I-General_Concept
(	O
BAHSIC	O
)	O
,	O
clustering	B-Algorithm
(	O
CLUHSIC	O
)	O
,	O
and	O
dimensionality	B-Algorithm
reduction	I-Algorithm
(	O
MUHSIC	O
)	O
.	O
</s>
<s>
where	O
denotes	O
the	O
element-wise	O
vector	O
product	O
,	O
is	O
the	O
set	O
of	O
nodes	O
connected	O
to	O
t	O
excluding	O
node	O
s	O
,	O
,	O
are	O
the	O
Gram	B-Algorithm
matrices	O
of	O
the	O
samples	O
from	O
variables	O
,	O
respectively	O
,	O
and	O
is	O
the	O
feature	O
matrix	O
for	O
the	O
samples	O
from	O
.	O
</s>
<s>
This	O
RKHS	O
function	O
representation	O
of	O
message-passing	O
updates	O
therefore	O
produces	O
an	O
efficient	O
belief	O
propagation	O
algorithm	O
in	O
which	O
the	O
potentials	O
are	O
nonparametric	B-General_Concept
functions	O
inferred	O
from	O
the	O
data	O
so	O
that	O
arbitrary	O
statistical	O
relationships	O
may	O
be	O
modeled	O
.	O
</s>
<s>
The	O
support	O
measure	O
machine	O
(	O
SMM	O
)	O
is	O
a	O
generalization	O
of	O
the	O
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
(	O
SVM	B-Algorithm
)	O
in	O
which	O
the	O
training	O
examples	O
are	O
probability	O
distributions	O
paired	O
with	O
labels	O
.	O
</s>
<s>
Under	O
certain	O
choices	O
of	O
the	O
embedding	O
kernel	O
,	O
the	O
SMM	O
applied	O
to	O
training	O
examples	O
is	O
equivalent	O
to	O
a	O
SVM	B-Algorithm
trained	O
on	O
samples	O
,	O
and	O
thus	O
the	O
SMM	O
can	O
be	O
viewed	O
as	O
a	O
flexible	O
SVM	B-Algorithm
in	O
which	O
a	O
different	O
data-dependent	O
kernel	O
(	O
specified	O
by	O
the	O
assumed	O
form	O
of	O
the	O
distribution	O
)	O
may	O
be	O
placed	O
on	O
each	O
training	O
point	O
.	O
</s>
<s>
The	O
goal	O
of	O
domain	B-General_Concept
adaptation	I-General_Concept
is	O
the	O
formulation	O
of	O
learning	O
algorithms	O
which	O
generalize	O
well	O
when	O
the	O
training	O
and	O
test	O
data	O
have	O
different	O
distributions	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
presence	O
of	O
conditional	O
shift	O
leads	O
to	O
an	O
ill-posed	B-Algorithm
problem	I-Algorithm
,	O
and	O
the	O
additional	O
assumption	O
that	O
changes	O
only	O
under	O
location-scale	O
(	O
LS	O
)	O
transformations	O
on	O
is	O
commonly	O
imposed	O
to	O
make	O
the	O
problem	O
tractable	O
.	O
</s>
<s>
DICA	O
thus	O
extracts	O
invariants	O
,	O
features	O
that	O
transfer	O
across	O
domains	O
,	O
and	O
may	O
be	O
viewed	O
as	O
a	O
generalization	O
of	O
many	O
popular	O
dimension-reduction	O
methods	O
such	O
as	O
kernel	B-Algorithm
principal	I-Algorithm
component	I-Algorithm
analysis	I-Algorithm
,	O
transfer	O
component	O
analysis	O
,	O
and	O
covariance	O
operator	O
inverse	O
regression	O
.	O
</s>
<s>
so	O
is	O
a	O
Gram	B-Algorithm
matrix	I-Algorithm
over	O
the	O
distributions	O
from	O
which	O
the	O
training	O
data	O
are	O
sampled	O
.	O
</s>
<s>
Finding	O
an	O
orthogonal	B-Algorithm
transform	I-Algorithm
onto	O
a	O
low-dimensional	O
subspace	O
B	O
(	O
in	O
the	O
feature	O
space	O
)	O
which	O
minimizes	O
the	O
distributional	O
variance	O
,	O
DICA	O
simultaneously	O
ensures	O
that	O
B	O
aligns	O
with	O
the	O
bases	O
of	O
a	O
central	O
subspace	O
C	O
for	O
which	O
becomes	O
independent	O
of	O
given	O
across	O
all	O
domains	O
.	O
</s>
<s>
Many	O
important	O
machine	O
learning	O
and	O
statistical	O
tasks	O
fit	O
into	O
this	O
framework	O
,	O
including	O
multi-instance	B-General_Concept
learning	I-General_Concept
,	O
and	O
point	O
estimation	O
problems	O
without	O
analytical	O
solution	O
(	O
such	O
as	O
hyperparameter	B-General_Concept
or	O
entropy	B-Algorithm
estimation	O
)	O
.	O
</s>
<s>
Distribution	O
regression	O
has	O
been	O
successfully	O
applied	O
for	O
example	O
in	O
supervised	O
entropy	B-Algorithm
learning	O
,	O
and	O
aerosol	O
prediction	O
using	O
multispectral	O
satellite	O
images	O
.	O
</s>
