<s>
In	O
statistical	O
theory	O
,	O
the	O
field	O
of	O
high-dimensional	B-Algorithm
statistics	I-Algorithm
studies	O
data	B-Application
whose	O
dimension	O
is	O
larger	O
than	O
typically	O
considered	O
in	O
classical	O
multivariate	O
analysis	O
.	O
</s>
<s>
The	O
area	O
arose	O
owing	O
to	O
the	O
emergence	O
of	O
many	O
modern	O
data	B-Application
sets	O
in	O
which	O
the	O
dimension	O
of	O
the	O
data	B-Application
vectors	O
may	O
be	O
comparable	O
to	O
,	O
or	O
even	O
larger	O
than	O
,	O
the	O
sample	O
size	O
,	O
so	O
that	O
justification	O
for	O
the	O
use	O
of	O
traditional	O
techniques	O
,	O
often	O
based	O
on	O
asymptotic	O
arguments	O
with	O
the	O
dimension	O
held	O
fixed	O
as	O
the	O
sample	O
size	O
increased	O
,	O
was	O
lacking	O
.	O
</s>
<s>
Given	O
independent	O
responses	O
,	O
with	O
corresponding	O
covariates	O
,	O
from	O
this	O
model	O
,	O
we	O
can	O
form	O
the	O
response	O
vector	O
,	O
and	O
design	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
When	O
and	O
the	O
design	B-Algorithm
matrix	I-Algorithm
has	O
full	O
column	O
rank	O
(	O
i.e.	O
</s>
<s>
When	O
,	O
it	O
is	O
known	B-General_Concept
that	O
.	O
</s>
<s>
However	O
,	O
overfitting	B-Error_Name
is	O
a	O
concern	O
when	O
is	O
of	O
comparable	O
magnitude	O
to	O
:	O
the	O
matrix	O
in	O
the	O
definition	O
of	O
may	O
become	O
ill-conditioned	B-Algorithm
,	O
with	O
a	O
small	O
minimum	O
eigenvalue	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
that	O
the	O
deterioration	O
in	O
estimation	O
performance	O
in	O
high	O
dimensions	O
observed	O
in	O
the	O
previous	O
paragraph	O
is	O
not	O
limited	O
to	O
the	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
estimator	O
.	O
</s>
<s>
In	O
fact	O
,	O
statistical	O
inference	O
in	O
high	O
dimensions	O
is	O
intrinsically	O
hard	O
,	O
a	O
phenomenon	O
known	B-General_Concept
as	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
,	O
and	O
it	O
can	O
be	O
shown	O
that	O
no	O
estimator	O
can	O
do	O
better	O
in	O
a	O
worst-case	O
sense	O
without	O
additional	O
information	O
(	O
see	O
Example	O
15.10	O
)	O
.	O
</s>
<s>
Nevertheless	O
,	O
the	O
situation	O
in	O
high-dimensional	B-Algorithm
statistics	I-Algorithm
may	O
not	O
be	O
hopeless	O
when	O
the	O
data	B-Application
possess	O
some	O
low-dimensional	O
structure	O
.	O
</s>
<s>
One	O
common	O
assumption	O
for	O
high-dimensional	O
linear	B-General_Concept
regression	I-General_Concept
is	O
that	O
the	O
vector	O
of	O
regression	B-General_Concept
coefficients	I-General_Concept
is	O
sparse	B-Algorithm
,	O
in	O
the	O
sense	O
that	O
most	O
coordinates	O
of	O
are	O
zero	O
.	O
</s>
<s>
Many	O
statistical	O
procedures	O
,	O
including	O
the	O
Lasso	B-Algorithm
,	O
have	O
been	O
proposed	O
to	O
fit	O
high-dimensional	O
linear	B-General_Concept
models	I-General_Concept
under	O
such	O
sparsity	B-Algorithm
assumptions	O
.	O
</s>
<s>
Another	O
example	O
of	O
a	O
high-dimensional	O
statistical	O
phenomenon	O
can	O
be	O
found	O
in	O
the	O
problem	O
of	O
covariance	B-General_Concept
matrix	I-General_Concept
estimation	I-General_Concept
.	O
</s>
<s>
Again	O
,	O
additional	O
low-dimensional	O
structure	O
is	O
needed	O
for	O
successful	O
covariance	B-General_Concept
matrix	I-General_Concept
estimation	I-General_Concept
in	O
high	O
dimensions	O
.	O
</s>
<s>
Examples	O
of	O
such	O
structures	O
include	O
sparsity	B-Algorithm
,	O
low	O
rankness	O
and	O
bandedness	B-Algorithm
.	O
</s>
<s>
Similar	O
remarks	O
apply	O
when	O
estimating	O
an	O
inverse	O
covariance	O
matrix	O
(	O
precision	B-General_Concept
matrix	I-General_Concept
)	O
.	O
</s>
<s>
From	O
an	O
applied	O
perspective	O
,	O
research	O
in	O
high-dimensional	B-Algorithm
statistics	I-Algorithm
was	O
motivated	O
by	O
the	O
realisation	O
that	O
advances	O
in	O
computing	O
technology	O
had	O
dramatically	O
increased	O
the	O
ability	O
to	O
collect	O
and	O
store	O
data	B-Application
,	O
and	O
that	O
traditional	O
statistical	O
techniques	O
such	O
as	O
those	O
described	O
in	O
the	O
examples	O
above	O
were	O
often	O
ill-equipped	O
to	O
handle	O
the	O
resulting	O
challenges	O
.	O
</s>
<s>
This	O
bias-variance	B-General_Concept
tradeoff	I-General_Concept
was	O
further	O
exploited	O
in	O
the	O
context	O
of	O
high-dimensional	O
linear	B-General_Concept
models	I-General_Concept
by	O
Hoerl	O
and	O
Kennard	O
in	O
1970	O
with	O
the	O
introduction	O
of	O
ridge	O
regression	O
.	O
</s>
<s>
Another	O
major	O
impetus	O
for	O
the	O
field	O
was	O
provided	O
by	O
Robert	O
Tibshirani	O
's	O
work	O
on	O
the	O
Lasso	B-Algorithm
in	O
1996	O
,	O
which	O
used	O
regularisation	O
to	O
achieve	O
simultaneous	O
model	O
selection	O
and	O
parameter	O
estimation	O
in	O
high-dimensional	O
sparse	B-Algorithm
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
The	O
following	O
are	O
examples	O
of	O
topics	O
that	O
have	O
received	O
considerable	O
attention	O
in	O
the	O
high-dimensional	B-Algorithm
statistics	I-Algorithm
literature	O
in	O
recent	O
years	O
:	O
</s>
<s>
Linear	B-General_Concept
models	I-General_Concept
in	O
high	O
dimensions	O
.	O
</s>
<s>
Linear	B-General_Concept
models	I-General_Concept
are	O
one	O
of	O
the	O
most	O
widely	O
used	O
tools	O
in	O
statistics	O
and	O
its	O
applications	O
.	O
</s>
<s>
As	O
such	O
,	O
sparse	B-Algorithm
linear	B-General_Concept
regression	I-General_Concept
is	O
one	O
of	O
the	O
most	O
well-studied	O
topics	O
in	O
high-dimensional	O
statistical	O
research	O
.	O
</s>
<s>
Building	O
upon	O
earlier	O
works	O
on	O
ridge	O
regression	O
and	O
the	O
Lasso	B-Algorithm
,	O
several	O
other	O
shrinkage	O
estimators	O
have	O
been	O
proposed	O
and	O
studied	O
in	O
this	O
and	O
related	O
problems	O
.	O
</s>
<s>
The	O
Dantzig	O
selector	O
,	O
which	O
minimises	O
the	O
maximum	O
covariate-residual	O
correlation	O
,	O
instead	O
of	O
the	O
residual	O
sum	O
of	O
squares	O
as	O
in	O
the	O
Lasso	B-Algorithm
,	O
subject	O
to	O
an	O
constraint	O
on	O
the	O
coefficients	O
.	O
</s>
<s>
Elastic	O
net	O
,	O
which	O
combines	O
regularisation	O
of	O
the	O
Lasso	B-Algorithm
with	O
regularisation	O
of	O
ridge	O
regression	O
to	O
allow	O
highly	O
correlated	O
covariates	O
to	O
be	O
simultaneously	O
selected	O
with	O
similar	O
regression	B-General_Concept
coefficients	I-General_Concept
.	O
</s>
<s>
The	O
Group	O
Lasso	B-Algorithm
,	O
which	O
allows	O
predefined	O
groups	O
of	O
covariates	O
to	O
be	O
selected	O
jointly	O
.	O
</s>
<s>
The	O
Fused	O
lasso	B-Algorithm
,	O
which	O
regularises	O
the	O
difference	O
between	O
nearby	O
coefficients	O
when	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
reflect	O
spatial	O
or	O
temporal	O
relationships	O
,	O
so	O
as	O
to	O
enforce	O
a	O
piecewise	O
constant	O
structure	O
.	O
</s>
<s>
High-dimensional	B-General_Concept
variable	I-General_Concept
selection	I-General_Concept
.	O
</s>
<s>
High-dimensional	O
covariance	O
and	O
precision	B-General_Concept
matrix	I-General_Concept
estimation	O
.	O
</s>
<s>
Sparse	B-Algorithm
principal	I-Algorithm
component	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Principal	B-Application
Component	I-Application
Analysis	I-Application
is	O
another	O
technique	O
that	O
breaks	O
down	O
in	O
high	O
dimensions	O
;	O
more	O
precisely	O
,	O
under	O
appropriate	O
conditions	O
,	O
the	O
leading	O
eigenvector	O
of	O
the	O
sample	O
covariance	O
matrix	O
is	O
an	O
inconsistent	O
estimator	O
of	O
its	O
population	O
counterpart	O
when	O
the	O
ratio	O
of	O
the	O
number	O
of	O
variables	O
to	O
the	O
number	O
of	O
observations	O
is	O
bounded	O
away	O
from	O
zero	O
.	O
</s>
<s>
Under	O
the	O
assumption	O
that	O
this	O
leading	O
eigenvector	O
is	O
sparse	B-Algorithm
(	O
which	O
can	O
aid	O
interpretability	O
)	O
,	O
consistency	O
can	O
be	O
restored	O
.	O
</s>
<s>
Linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
cannot	O
be	O
used	O
when	O
,	O
because	O
the	O
sample	O
covariance	O
matrix	O
is	O
singular	O
.	O
</s>
<s>
Alternative	O
approaches	O
have	O
been	O
proposed	O
based	O
on	O
naive	B-General_Concept
Bayes	I-General_Concept
,	O
feature	B-General_Concept
selection	I-General_Concept
and	O
random	B-Architecture
projections	I-Architecture
.	O
</s>
<s>
Graphical	B-Algorithm
models	I-Algorithm
for	I-Algorithm
high-dimensional	I-Algorithm
data	I-Algorithm
.	O
</s>
<s>
Graphical	B-Algorithm
models	I-Algorithm
are	O
used	O
to	O
encode	O
the	O
conditional	O
dependence	O
structure	O
between	O
different	O
variables	O
.	O
</s>
<s>
Under	O
a	O
Gaussianity	O
assumption	O
,	O
the	O
problem	O
reduces	O
to	O
that	O
of	O
estimating	O
a	O
sparse	B-Algorithm
precision	B-General_Concept
matrix	I-General_Concept
,	O
discussed	O
above	O
.	O
</s>
