<s>
The	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
refers	O
to	O
various	O
phenomena	O
that	O
arise	O
when	O
analyzing	O
and	O
organizing	O
data	O
in	O
high-dimensional	O
spaces	O
that	O
do	O
not	O
occur	O
in	O
low-dimensional	O
settings	O
such	O
as	O
the	O
three-dimensional	O
physical	O
space	O
of	O
everyday	O
experience	O
.	O
</s>
<s>
The	O
expression	O
was	O
coined	O
by	O
Richard	O
E	O
.	O
Bellman	O
when	O
considering	O
problems	O
in	O
dynamic	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Dimensionally	O
cursed	O
phenomena	O
occur	O
in	O
domains	O
such	O
as	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
sampling	O
,	O
combinatorics	O
,	O
machine	O
learning	O
,	O
data	B-Application
mining	I-Application
and	O
databases	O
.	O
</s>
<s>
For	O
example	O
,	O
102	O
=	O
100	O
evenly	O
spaced	O
sample	O
points	O
suffice	O
to	O
sample	O
a	O
unit	O
interval	O
(	O
a	O
"	O
1-dimensional	O
cube	O
"	O
)	O
with	O
no	O
more	O
than	O
10−2	O
=	O
0.01	O
distance	O
between	O
points	O
;	O
an	O
equivalent	O
sampling	O
of	O
a	O
10-dimensional	O
unit	O
hypercube	B-Operating_System
with	O
a	O
lattice	O
that	O
has	O
a	O
spacing	O
of	O
10−2	O
=	O
0.01	O
between	O
adjacent	O
points	O
would	O
require	O
1020	O
=	O
[( 	O
102	O
)	O
10 ]	O
sample	O
points	O
.	O
</s>
<s>
In	O
general	O
,	O
with	O
a	O
spacing	O
distance	O
of	O
10−n	O
the	O
10-dimensional	O
hypercube	B-Operating_System
appears	O
to	O
be	O
a	O
factor	O
of	O
10n( 10−1	O
)	O
=	O
[( 	O
10n	O
)	O
10/	O
( 10n	O
)	O
]	O
"	O
larger	O
"	O
than	O
the	O
1-dimensional	O
hypercube	B-Operating_System
,	O
which	O
is	O
the	O
unit	O
interval	O
.	O
</s>
<s>
In	O
the	O
above	O
example	O
n	O
=	O
2	O
:	O
when	O
using	O
a	O
sampling	O
distance	O
of	O
0.01	O
the	O
10-dimensional	O
hypercube	B-Operating_System
appears	O
to	O
be	O
1018	O
"	O
larger	O
"	O
than	O
the	O
unit	O
interval	O
.	O
</s>
<s>
When	O
solving	O
dynamic	B-Algorithm
optimization	I-Algorithm
problems	O
by	O
numerical	O
backward	O
induction	O
,	O
the	O
objective	O
function	O
must	O
be	O
computed	O
for	O
each	O
combination	O
of	O
values	O
.	O
</s>
<s>
In	O
machine	O
learning	O
and	O
insofar	O
as	O
predictive	O
performance	O
is	O
concerned	O
,	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
is	O
used	O
interchangeably	O
with	O
the	O
peaking	O
phenomenon	O
,	O
which	O
is	O
also	O
known	O
as	O
Hughes	O
phenomenon	O
.	O
</s>
<s>
This	O
phenomenon	O
states	O
that	O
with	O
a	O
fixed	O
number	O
of	O
training	O
samples	O
,	O
the	O
average	O
(	O
expected	O
)	O
predictive	O
power	O
of	O
a	O
classifier	B-General_Concept
or	O
regressor	O
first	O
increases	O
as	O
the	O
number	O
of	O
dimensions	O
or	O
features	O
used	O
is	O
increased	O
but	O
beyond	O
a	O
certain	O
dimensionality	O
it	O
starts	O
deteriorating	O
instead	O
of	O
improving	O
steadily	O
.	O
</s>
<s>
Nevertheless	O
,	O
in	O
the	O
context	O
of	O
a	O
simple	O
classifier	B-General_Concept
(	O
linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
in	O
the	O
multivariate	O
Gaussian	O
model	O
under	O
the	O
assumption	O
of	O
a	O
common	O
known	O
covariance	O
matrix	O
)	O
Zollanvari	O
et	O
al	O
.	O
</s>
<s>
showed	O
both	O
analytically	O
and	O
empirically	O
that	O
as	O
long	O
as	O
the	O
relative	O
cumulative	O
efficacy	O
of	O
an	O
additional	O
feature	O
set	O
(	O
with	O
respect	O
to	O
features	O
that	O
are	O
already	O
part	O
of	O
the	O
classifier	B-General_Concept
)	O
is	O
greater	O
(	O
or	O
less	O
)	O
than	O
the	O
size	O
of	O
this	O
additional	O
feature	O
set	O
,	O
the	O
expected	O
error	O
of	O
the	O
classifier	B-General_Concept
constructed	O
using	O
these	O
additional	O
features	O
will	O
be	O
less	O
(	O
or	O
greater	O
)	O
than	O
the	O
expected	O
error	O
of	O
the	O
classifier	B-General_Concept
constructed	O
without	O
them	O
.	O
</s>
<s>
+	O
Genetic	O
mutations	O
in	O
individuals	O
data	B-General_Concept
set	I-General_Concept
Individual	O
name	O
Gene	O
1	O
Gene	O
2	O
...	O
Gene	O
2000	O
Individual	O
1	O
1	O
0	O
...	O
1	O
...	O
...	O
...	O
...	O
...	O
</s>
<s>
In	O
data	B-Application
mining	I-Application
,	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
refers	O
to	O
a	O
data	B-General_Concept
set	I-General_Concept
with	O
too	O
many	O
features	O
.	O
</s>
<s>
A	O
data	B-Application
mining	I-Application
application	O
to	O
this	O
data	B-General_Concept
set	I-General_Concept
may	O
be	O
finding	O
the	O
correlation	O
between	O
specific	O
genetic	O
mutations	O
and	O
creating	O
a	O
classification	B-General_Concept
algorithm	O
such	O
as	O
a	O
decision	B-Algorithm
tree	I-Algorithm
to	O
determine	O
whether	O
an	O
individual	O
has	O
cancer	O
or	O
not	O
.	O
</s>
<s>
A	O
common	O
practice	O
of	O
data	B-Application
mining	I-Application
in	O
this	O
domain	O
would	O
be	O
to	O
create	O
association	B-Algorithm
rules	I-Algorithm
between	O
genetic	O
mutations	O
that	O
lead	O
to	O
the	O
development	O
of	O
cancers	O
.	O
</s>
<s>
As	O
we	O
can	O
see	O
from	O
the	O
permutation	O
table	O
above	O
,	O
one	O
of	O
the	O
major	O
problems	O
data	B-Application
miners	I-Application
face	O
regarding	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
is	O
that	O
the	O
space	O
of	O
possible	O
parameter	O
values	O
grows	O
exponentially	O
or	O
factorially	O
as	O
the	O
number	O
of	O
features	O
in	O
the	O
data	B-General_Concept
set	I-General_Concept
grows	O
.	O
</s>
<s>
Another	O
problem	O
data	B-Application
miners	I-Application
may	O
face	O
when	O
dealing	O
with	O
too	O
many	O
features	O
is	O
the	O
notion	O
that	O
the	O
number	O
of	O
false	O
predictions	O
or	O
classifications	O
tend	O
to	O
increase	O
as	O
the	O
number	O
of	O
features	O
grow	O
in	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
In	O
terms	O
of	O
the	O
classification	B-General_Concept
problem	O
discussed	O
above	O
,	O
keeping	O
every	O
data	O
point	O
could	O
lead	O
to	O
a	O
higher	O
number	O
of	O
false	O
positives	O
and	O
false	O
negatives	O
in	O
the	O
model	O
.	O
</s>
<s>
This	O
problem	O
is	O
up	O
to	O
the	O
data	B-Application
miner	I-Application
to	O
solve	O
,	O
and	O
there	O
is	O
no	O
universal	O
solution	O
.	O
</s>
<s>
The	O
first	O
step	O
any	O
data	B-Application
miner	I-Application
should	O
take	O
is	O
to	O
explore	O
the	O
data	O
,	O
in	O
an	O
attempt	O
to	O
gain	O
an	O
understanding	O
of	O
how	O
it	O
can	O
be	O
used	O
to	O
solve	O
the	O
problem	O
.	O
</s>
<s>
One	O
must	O
first	O
understand	O
what	O
the	O
data	O
means	O
,	O
and	O
what	O
they	O
are	O
trying	O
to	O
discover	O
before	O
they	O
can	O
decide	O
if	O
anything	O
must	O
be	O
removed	O
from	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
Then	O
they	O
can	O
create	O
or	O
use	O
a	O
feature	B-General_Concept
selection	I-General_Concept
or	O
dimensionality	B-Algorithm
reduction	I-Algorithm
algorithm	I-Algorithm
to	O
remove	O
samples	O
or	O
features	O
from	O
the	O
data	B-General_Concept
set	I-General_Concept
if	O
they	O
deem	O
it	O
necessary	O
.	O
</s>
<s>
One	O
example	O
of	O
such	O
methods	O
is	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
method	O
,	O
used	O
to	O
remove	O
outliers	O
in	O
a	O
data	B-General_Concept
set	I-General_Concept
by	O
calculating	O
the	O
standard	O
deviation	O
of	O
a	O
feature	O
or	O
occurrence	O
.	O
</s>
<s>
As	O
the	O
dimension	O
of	O
the	O
space	O
increases	O
,	O
the	O
hypersphere	O
becomes	O
an	O
insignificant	O
volume	O
relative	O
to	O
that	O
of	O
the	O
hypercube	B-Operating_System
.	O
</s>
<s>
In	O
this	O
sense	O
when	O
points	O
are	O
uniformly	O
generated	O
in	O
a	O
high-dimensional	O
hypercube	B-Operating_System
,	O
almost	O
all	O
points	O
are	O
much	O
farther	O
than	O
units	O
away	O
from	O
the	O
centre	O
.	O
</s>
<s>
Thus	O
,	O
when	O
uniformly	O
generating	O
points	O
in	O
high	O
dimensions	O
,	O
both	O
the	O
"	O
middle	O
"	O
of	O
the	O
hypercube	B-Operating_System
,	O
and	O
the	O
corners	O
are	O
empty	O
,	O
and	O
all	O
the	O
volume	O
is	O
concentrated	O
near	O
the	O
surface	O
of	O
a	O
sphere	O
of	O
"	O
intermediate	O
"	O
radius	O
.	O
</s>
<s>
When	O
attributes	O
are	O
correlated	O
,	O
data	O
can	O
become	O
easier	O
and	O
provide	O
higher	O
distance	O
contrast	O
and	O
the	O
signal-to-noise	O
ratio	O
was	O
found	O
to	O
play	O
an	O
important	O
role	O
,	O
thus	O
feature	B-General_Concept
selection	I-General_Concept
should	O
be	O
used	O
.	O
</s>
<s>
The	O
effect	O
complicates	O
nearest	B-Algorithm
neighbor	I-Algorithm
search	I-Algorithm
in	O
high	O
dimensional	O
space	O
.	O
</s>
<s>
Another	O
effect	O
of	O
high	O
dimensionality	O
on	O
distance	O
functions	O
concerns	O
k-nearest	B-General_Concept
neighbor	I-General_Concept
(	O
k-NN	B-General_Concept
)	O
graphs	O
constructed	O
from	O
a	O
data	B-General_Concept
set	I-General_Concept
using	O
a	O
distance	O
function	O
.	O
</s>
<s>
As	O
the	O
dimension	O
increases	O
,	O
the	O
indegree	O
distribution	O
of	O
the	O
k-NN	B-General_Concept
digraph	O
becomes	O
skewed	B-General_Concept
with	O
a	O
peak	O
on	O
the	O
right	O
because	O
of	O
the	O
emergence	O
of	O
a	O
disproportionate	O
number	O
of	O
hubs	O
,	O
that	O
is	O
,	O
data-points	O
that	O
appear	O
in	O
many	O
more	O
k-NN	B-General_Concept
lists	O
of	O
other	O
data-points	O
than	O
the	O
average	O
.	O
</s>
<s>
This	O
phenomenon	O
can	O
have	O
a	O
considerable	O
impact	O
on	O
various	O
techniques	O
for	O
classification	B-General_Concept
(	O
including	O
the	O
k-NN	B-General_Concept
classifier	B-General_Concept
)	O
,	O
semi-supervised	B-General_Concept
learning	I-General_Concept
,	O
and	O
clustering	B-Algorithm
,	O
and	O
it	O
also	O
affects	O
information	B-Library
retrieval	I-Library
.	O
</s>
<s>
identified	O
the	O
following	O
problems	O
when	O
searching	O
for	O
anomalies	B-Algorithm
in	O
high-dimensional	O
data	O
:	O
</s>
<s>
Surprisingly	O
and	O
despite	O
the	O
expected	O
"	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
"	O
difficulties	O
,	O
common-sense	O
heuristics	O
based	O
on	O
the	O
most	O
straightforward	O
methods	O
"	O
can	O
yield	O
results	O
which	O
are	O
almost	O
surely	O
optimal	O
"	O
for	O
high-dimensional	O
problems	O
.	O
</s>
<s>
Donoho	O
in	O
his	O
"	O
Millennium	O
manifesto	O
"	O
clearly	O
explained	O
why	O
the	O
"	O
blessing	O
of	O
dimensionality	O
"	O
will	O
form	O
a	O
basis	O
of	O
future	O
data	B-Application
mining	I-Application
.	O
</s>
<s>
Moreover	O
,	O
this	O
linear	O
functional	O
can	O
be	O
selected	O
in	O
the	O
form	O
of	O
the	O
simplest	O
linear	O
Fisher	B-General_Concept
discriminant	I-General_Concept
.	O
</s>
<s>
"	O
The	O
blessing	O
of	O
dimensionality	O
and	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
are	O
two	O
sides	O
of	O
the	O
same	O
coin.	O
"	O
</s>
<s>
noted	O
that	O
while	O
the	O
typical	O
formalizations	O
of	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
affect	O
i.i.d.	O
</s>
