<s>
Clustering	B-Algorithm
high-dimensional	I-Algorithm
data	I-Algorithm
is	O
the	O
cluster	B-Algorithm
analysis	I-Algorithm
of	O
data	O
with	O
anywhere	O
from	O
a	O
few	O
dozen	O
to	O
many	O
thousands	O
of	O
dimensions	O
.	O
</s>
<s>
Such	O
high-dimensional	O
spaces	O
of	O
data	O
are	O
often	O
encountered	O
in	O
areas	O
such	O
as	O
medicine	O
,	O
where	O
DNA	O
microarray	O
technology	O
can	O
produce	O
many	O
measurements	O
at	O
once	O
,	O
and	O
the	O
clustering	O
of	O
text	B-General_Concept
documents	I-General_Concept
,	O
where	O
,	O
if	O
a	O
word-frequency	O
vector	O
is	O
used	O
,	O
the	O
number	O
of	O
dimensions	O
equals	O
the	O
size	B-General_Concept
of	I-General_Concept
the	I-General_Concept
vocabulary	I-General_Concept
.	O
</s>
<s>
This	O
problem	O
is	O
known	O
as	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
An	O
overall	O
different	O
approach	O
is	O
to	O
find	O
clusters	O
based	O
on	O
pattern	O
in	O
the	O
data	O
matrix	O
,	O
often	O
referred	O
to	O
as	O
biclustering	B-Algorithm
,	O
which	O
is	O
a	O
technique	O
frequently	O
utilized	O
in	O
bioinformatics	O
.	O
</s>
<s>
Hence	O
,	O
subspace	O
clustering	B-Algorithm
algorithms	I-Algorithm
utilize	O
some	O
kind	O
of	O
heuristic	B-Algorithm
to	O
remain	O
computationally	O
feasible	O
,	O
at	O
the	O
risk	O
of	O
producing	O
inferior	O
results	O
.	O
</s>
<s>
association	B-Algorithm
rules	I-Algorithm
)	O
can	O
be	O
used	O
to	O
build	O
higher-dimensional	O
subspaces	O
only	O
by	O
combining	O
lower-dimensional	O
ones	O
,	O
as	O
any	O
subspace	O
T	O
containing	O
a	O
cluster	O
,	O
will	O
result	O
in	O
a	O
full	O
space	O
S	O
also	O
to	O
contain	O
that	O
cluster	O
(	O
i.e.	O
</s>
<s>
S	O
⊆	O
T	O
)	O
,	O
an	O
approach	O
taken	O
by	O
most	O
of	O
the	O
traditional	O
algorithms	O
such	O
as	O
CLIQUE	O
,	O
SUBCLU	B-Algorithm
.	O
</s>
<s>
The	O
general	O
approach	O
is	O
to	O
use	O
a	O
special	O
distance	O
function	O
together	O
with	O
a	O
regular	O
clustering	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
the	O
figure	O
above	O
,	O
the	O
cluster	O
might	O
be	O
found	O
using	O
DBSCAN	B-Algorithm
with	O
a	O
distance	O
function	O
that	O
places	O
less	O
emphasis	O
on	O
the	O
-axis	O
and	O
thus	O
exaggerates	O
the	O
low	O
difference	O
in	O
the	O
-axis	O
sufficiently	O
enough	O
to	O
group	O
the	O
points	O
into	O
a	O
cluster	O
.	O
</s>
<s>
PROCLUS	O
uses	O
a	O
similar	O
approach	O
with	O
a	O
k-medoid	B-Algorithm
clustering	O
.	O
</s>
<s>
The	O
algorithm	O
then	O
proceeds	O
as	O
the	O
regular	O
PAM	B-Algorithm
algorithm	O
.	O
</s>
<s>
If	O
the	O
distance	O
function	O
weights	O
attributes	O
differently	O
,	O
but	O
never	O
with	O
0	O
(	O
and	O
hence	O
never	O
drops	O
irrelevant	O
attributes	O
)	O
,	O
the	O
algorithm	O
is	O
called	O
a	O
"	O
soft	O
"	O
-projected	O
clustering	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Typical	O
projection-methods	O
like	O
t-distributed	B-Algorithm
stochastic	I-Algorithm
neighbor	I-Algorithm
embedding	I-Algorithm
(	O
t-SNE	B-Algorithm
)	O
,	O
or	O
neighbor	O
retrieval	O
visualizer	O
(	O
NerV	O
)	O
are	O
used	O
to	O
project	O
data	O
explicitly	O
into	O
two	O
dimensions	O
disregarding	O
the	O
subspaces	O
of	O
higher	O
dimension	O
than	O
two	O
and	O
preserving	O
only	O
relevant	O
neighborhoods	O
in	O
high-dimensional	O
data	O
.	O
</s>
<s>
In	O
the	O
next	O
step	O
,	O
the	O
Delaunay	B-Algorithm
graph	I-Algorithm
between	O
the	O
projected	O
points	O
is	O
calculated	O
,	O
and	O
each	O
vertex	O
between	O
two	O
projected	O
points	O
is	O
weighted	O
with	O
the	O
high-dimensional	O
distance	O
between	O
the	O
corresponding	O
high-dimensional	O
data	O
points	O
.	O
</s>
<s>
Thereafter	O
the	O
shortest	O
path	O
between	O
every	O
pair	O
of	O
points	O
is	O
computed	O
using	O
the	O
Dijkstra	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
An	O
example	O
is	O
FIRES	O
,	O
which	O
is	O
from	O
its	O
basic	O
approach	O
a	O
subspace	O
clustering	B-Algorithm
algorithm	I-Algorithm
,	O
but	O
uses	O
a	O
heuristic	B-Algorithm
too	O
aggressive	O
to	O
credibly	O
produce	O
all	O
subspace	O
clusters	O
.	O
</s>
<s>
Another	O
hybrid	O
approach	O
is	O
to	O
include	O
a	O
human-into-the-algorithmic-loop	O
:	O
Human	O
domain	O
expertise	O
can	O
help	O
to	O
reduce	O
an	O
exponential	O
search	O
space	O
through	O
heuristic	B-Algorithm
selection	O
of	O
samples	O
.	O
</s>
<s>
Another	O
type	O
of	O
subspaces	O
is	O
considered	O
in	O
Correlation	B-Algorithm
clustering	I-Algorithm
(	O
Data	O
Mining	O
)	O
.	O
</s>
