<s>
Neighbourhood	B-General_Concept
components	I-General_Concept
analysis	I-General_Concept
is	O
a	O
supervised	B-General_Concept
learning	I-General_Concept
method	O
for	O
classifying	B-General_Concept
multivariate	B-General_Concept
data	I-General_Concept
into	O
distinct	O
classes	O
according	O
to	O
a	O
given	O
distance	O
metric	O
over	O
the	O
data	O
.	O
</s>
<s>
Functionally	O
,	O
it	O
serves	O
the	O
same	O
purposes	O
as	O
the	O
K-nearest	B-General_Concept
neighbors	I-General_Concept
algorithm	I-General_Concept
,	O
and	O
makes	O
direct	O
use	O
of	O
a	O
related	O
concept	O
termed	O
stochastic	O
nearest	O
neighbours	O
.	O
</s>
<s>
Neighbourhood	B-General_Concept
components	I-General_Concept
analysis	I-General_Concept
aims	O
at	O
"	O
learning	O
"	O
a	O
distance	O
metric	O
by	O
finding	O
a	O
linear	O
transformation	O
of	O
input	O
data	O
such	O
that	O
the	O
average	O
leave-one-out	O
(	O
LOO	O
)	O
classification	O
performance	O
is	O
maximized	O
in	O
the	O
transformed	O
space	O
.	O
</s>
<s>
The	O
key	O
insight	O
to	O
the	O
algorithm	O
is	O
that	O
a	O
matrix	O
corresponding	O
to	O
the	O
transformation	O
can	O
be	O
found	O
by	O
defining	O
a	O
differentiable	O
objective	O
function	O
for	O
,	O
followed	O
by	O
use	O
of	O
an	O
iterative	O
solver	O
such	O
as	O
conjugate	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
We	O
can	O
resolve	O
this	O
difficulty	O
by	O
using	O
an	O
approach	O
inspired	O
by	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
We	O
define	O
these	O
using	O
a	O
softmax	B-Algorithm
function	I-Algorithm
of	O
the	O
squared	O
Euclidean	O
distance	O
between	O
a	O
given	O
LOO-classification	O
point	O
and	O
each	O
other	O
point	O
in	O
the	O
transformed	O
space	O
:	O
</s>
<s>
The	O
probability	O
of	O
correctly	O
classifying	B-General_Concept
data	O
point	O
is	O
the	O
probability	O
of	O
classifying	B-General_Concept
the	O
points	O
of	O
each	O
of	O
its	O
neighbours	O
with	O
the	O
same	O
class	O
:	O
</s>
<s>
where	O
is	O
the	O
probability	O
of	O
classifying	B-General_Concept
neighbour	O
of	O
point	O
.	O
</s>
<s>
Thus	O
the	O
predicted	O
class	O
is	O
an	O
affine	O
combination	O
of	O
the	O
classes	O
of	O
every	O
other	O
point	O
,	O
weighted	O
by	O
the	O
softmax	B-Algorithm
function	I-Algorithm
for	O
each	O
where	O
is	O
now	O
the	O
entire	O
transformed	O
data	O
set	O
.	O
</s>
<s>
Obtaining	O
a	O
gradient	O
for	O
means	O
that	O
it	O
can	O
be	O
found	O
with	O
an	O
iterative	O
solver	O
such	O
as	O
conjugate	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
Neighbourhood	B-General_Concept
components	I-General_Concept
analysis	I-General_Concept
was	O
developed	O
by	O
Jacob	O
Goldberger	O
,	O
Sam	O
Roweis	O
,	O
Ruslan	O
Salakhudinov	O
,	O
and	O
Geoff	O
Hinton	O
at	O
the	O
University	O
of	O
Toronto	O
's	O
department	O
of	O
computer	O
science	O
in	O
2004	O
.	O
</s>
