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( T o p )
1
D e f i n i t i o n
T o g g l e D e f i n i t i o n s u b s e c t i o n
1 . 1
A r g m i n
2
E x a m p l e s a n d p r o p e r t i e s
3
S e e a l s o
4
N o t e s
5
R e f e r e n c e s
6
E x t e r n a l l i n k s
T o g g l e t h e t a b l e o f c o n t e n t s
A r g m a x
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Inputs at which function values are highest
As an example, both unnormalised and normalised sinc functions above have
argmax
{\displaystyle \operatorname {argmax} }
of {0} because both attain their global maximum value of 1 at x = 0. The unnormalised sinc function (red) has arg min of {−4.49, 4.49}, approximately, because it has 2 global minimum values of approximately −0.217 at x = ±4.49. However, the normalised sinc function (blue) has arg min of {−1.43, 1.43}, approximately, because their global minima occur at x = ±1.43, even though the minimum value is the same.[1]
In mathematics , the arguments of the maxima (abbreviated arg max or argmax ) and arguments of the minima (abbreviated arg min or argmin ) are the input points at which a function output value is maximized and minimized , respectively.[note 1] While the arguments are defined over the domain of a function , the output is part of its codomain .
Definition
[ edit ]
Given an arbitrary set
X
{\displaystyle X}
,a totally ordered set
Y
{\displaystyle Y}
, and a function,
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
, the
argmax
{\displaystyle \operatorname {argmax} }
over some subset
S
{\displaystyle S}
of
X
{\displaystyle X}
is defined by
argmax
S
f
:=
a
r
g
m
a
x
x
∈
S
f
(
x
)
:=
{
x
∈
S
:
f
(
s
)
≤
f
(
x
)
for all
s
∈
S
}
.
{\displaystyle \operatorname {argmax} _{S}f:={\underset {x\in S}{\operatorname {arg\,max} }}\,f(x ):=\{x\in S~:~f(s )\leq f(x ){\text{ for all }}s\in S\}.}
If
S
=
X
{\displaystyle S=X}
or
S
{\displaystyle S}
is clear from the context, then
S
{\displaystyle S}
is often left out, as in
a
r
g
m
a
x
x
f
(
x
)
:=
{
x
:
f
(
s
)
≤
f
(
x
)
for all
s
∈
X
}
.
{\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,f(x ):=\{x~:~f(s )\leq f(x ){\text{ for all }}s\in X\}.}
In other words,
argmax
{\displaystyle \operatorname {argmax} }
is the set of points
x
{\displaystyle x}
for which
f
(
x
)
{\displaystyle f(x )}
attains the function's largest value (if it exists).
Argmax
{\displaystyle \operatorname {Argmax} }
may be the empty set , a singleton , or contain multiple elements.
In the fields of convex analysis and variational analysis , a slightly different definition is used in the special case where
Y
=
[
−
∞
,
∞
]
=
R
∪
{
±
∞
}
{\displaystyle Y=[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}
are the extended real numbers . In this case, if
f
{\displaystyle f}
is identically equal to
∞
{\displaystyle \infty }
on
S
{\displaystyle S}
then
argmax
S
f
:=
∅
{\displaystyle \operatorname {argmax} _{S}f:=\varnothing }
(that is,
argmax
S
∞
:=
∅
{\displaystyle \operatorname {argmax} _{S}\infty :=\varnothing }
) and otherwise
argmax
S
f
{\displaystyle \operatorname {argmax} _{S}f}
is defined as above, where in this case
argmax
S
f
{\displaystyle \operatorname {argmax} _{S}f}
can also be written as:
argmax
S
f
:=
{
x
∈
S
:
f
(
x
)
=
sup
S
f
}
{\displaystyle \operatorname {argmax} _{S}f:=\left\{x\in S~:~f(x )=\sup {}_{S}f\right\}}
where it is emphasized that this equality involving
sup
S
f
{\displaystyle \sup {}_{S}f}
holds only when
f
{\displaystyle f}
is not identically
∞
{\displaystyle \infty }
on
S
{\displaystyle S}
.
Arg min
[ edit ]
The notion of
argmin
{\displaystyle \operatorname {argmin} }
(or
a
r
g
m
i
n
{\displaystyle \operatorname {arg\,min} }
), which stands for argument of the minimum , is defined analogously. For instance,
a
r
g
m
i
n
x
∈
S
f
(
x
)
:=
{
x
∈
S
:
f
(
s
)
≥
f
(
x
)
for all
s
∈
S
}
{\displaystyle {\underset {x\in S}{\operatorname {arg\,min} }}\,f(x ):=\{x\in S~:~f(s )\geq f(x ){\text{ for all }}s\in S\}}
are points
x
{\displaystyle x}
for which
f
(
x
)
{\displaystyle f(x )}
attains its smallest value. It is the complementary operator of
a
r
g
m
a
x
{\displaystyle \operatorname {arg\,max} }
.
In the special case where
Y
=
[
−
∞
,
∞
]
=
R
∪
{
±
∞
}
{\displaystyle Y=[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}
are the extended real numbers , if
f
{\displaystyle f}
is identically equal to
−
∞
{\displaystyle -\infty }
on
S
{\displaystyle S}
then
argmin
S
f
:=
∅
{\displaystyle \operatorname {argmin} _{S}f:=\varnothing }
(that is,
argmin
S
−
∞
:=
∅
{\displaystyle \operatorname {argmin} _{S}-\infty :=\varnothing }
) and otherwise
argmin
S
f
{\displaystyle \operatorname {argmin} _{S}f}
is defined as above and moreover, in this case (of
f
{\displaystyle f}
not identically equal to
−
∞
{\displaystyle -\infty }
) it also satisfies:
argmin
S
f
:=
{
x
∈
S
:
f
(
x
)
=
inf
S
f
}
.
{\displaystyle \operatorname {argmin} _{S}f:=\left\{x\in S~:~f(x )=\inf {}_{S}f\right\}.}
Examples and properties
[ edit ]
For example, if
f
(
x
)
{\displaystyle f(x )}
is
1
−
|
x
|
,
{\displaystyle 1-|x|,}
then
f
{\displaystyle f}
attains its maximum value of
1
{\displaystyle 1}
only at the point
x
=
0.
{\displaystyle x=0.}
Thus
a
r
g
m
a
x
x
(
1
−
|
x
|
)
=
{
0
}
.
{\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,(1-|x|)=\{0\}.}
The
argmax
{\displaystyle \operatorname {argmax} }
operator is different from the
max
{\displaystyle \max }
operator. The
max
{\displaystyle \max }
operator, when given the same function, returns the maximum value of the function instead of the point or points that cause that function to reach that value; in other words
max
x
f
(
x
)
{\displaystyle \max _{x}f(x )}
is the element in
{
f
(
x
)
:
f
(
s
)
≤
f
(
x
)
for all
s
∈
S
}
.
{\displaystyle \{f(x )~:~f(s )\leq f(x ){\text{ for all }}s\in S\}.}
Like
argmax
,
{\displaystyle \operatorname {argmax} ,}
max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike
argmax
,
{\displaystyle \operatorname {argmax} ,}
max
{\displaystyle \operatorname {max} }
may not contain multiple elements:[note 2] for example, if
f
(
x
)
{\displaystyle f(x )}
is
4
x
2
−
x
4
,
{\displaystyle 4x^{2}-x^{4},}
then
a
r
g
m
a
x
x
(
4
x
2
−
x
4
)
=
{
−
2
,
2
}
,
{\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,\left(4x^{2}-x^{4}\right)=\left\{-{\sqrt {2}},{\sqrt {2}}\right\},}
but
max
x
(
4
x
2
−
x
4
)
=
{
4
}
{\displaystyle {\underset {x}{\operatorname {max} }}\,\left(4x^{2}-x^{4}\right)=\{4\}}
because the function attains the same value at every element of
argmax
.
{\displaystyle \operatorname {argmax} .}
Equivalently, if
M
{\displaystyle M}
is the maximum of
f
,
{\displaystyle f,}
then the
argmax
{\displaystyle \operatorname {argmax} }
is the level set of the maximum:
a
r
g
m
a
x
x
f
(
x
)
=
{
x
:
f
(
x
)
=
M
}
=:
f
−
1
(
M
)
.
{\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,f(x )=\{x~:~f(x )=M\}=:f^{-1}(M ).}
We can rearrange to give the simple identity[note 3]
f
(
a
r
g
m
a
x
x
f
(
x
)
)
=
max
x
f
(
x
)
.
{\displaystyle f\left({\underset {x}{\operatorname {arg\,max} }}\,f(x )\right)=\max _{x}f(x ).}
If the maximum is reached at a single point then this point is often referred to as the
argmax
,
{\displaystyle \operatorname {argmax} ,}
and
argmax
{\displaystyle \operatorname {argmax} }
is considered a point, not a set of points. So, for example,
a
r
g
m
a
x
x
∈
R
(
x
(
10
−
x
)
)
=
5
{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,(x(10-x))=5}
(rather than the singleton set
{
5
}
{\displaystyle \{5\}}
), since the maximum value of
x
(
10
−
x
)
{\displaystyle x(10-x)}
is
25
,
{\displaystyle 25,}
which occurs for
x
=
5.
{\displaystyle x=5.}
[note 4] However, in case the maximum is reached at many points,
argmax
{\displaystyle \operatorname {argmax} }
needs to be considered a set of points.
For example
a
r
g
m
a
x
x
∈
[
0
,
4
π
]
cos
(
x
)
=
{
0
,
2
π
,
4
π
}
{\displaystyle {\underset {x\in [0,4\pi ]}{\operatorname {arg\,max} }}\,\cos(x )=\{0,2\pi ,4\pi \}}
because the maximum value of
cos
x
{\displaystyle \cos x}
is
1
,
{\displaystyle 1,}
which occurs on this interval for
x
=
0
,
2
π
{\displaystyle x=0,2\pi }
or
4
π
.
{\displaystyle 4\pi .}
On the whole real line
a
r
g
m
a
x
x
∈
R
cos
(
x
)
=
{
2
k
π
:
k
∈
Z
}
,
{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,\cos(x )=\left\{2k\pi ~:~k\in \mathbb {Z} \right\},}
so an infinite set.
Functions need not in general attain a maximum value, and hence the
argmax
{\displaystyle \operatorname {argmax} }
is sometimes the empty set ; for example,
a
r
g
m
a
x
x
∈
R
x
3
=
∅
,
{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,x^{3}=\varnothing ,}
since
x
3
{\displaystyle x^{3}}
is unbounded on the real line. As another example,
a
r
g
m
a
x
x
∈
R
arctan
(
x
)
=
∅
,
{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,\arctan(x )=\varnothing ,}
although
arctan
{\displaystyle \arctan }
is bounded by
±
π
/
2.
{\displaystyle \pm \pi /2.}
However, by the extreme value theorem , a continuous real-valued function on a closed interval has a maximum, and thus a nonempty
argmax
.
{\displaystyle \operatorname {argmax} .}
See also
[ edit ]
Notes
[ edit ]
^ Due to the anti-symmetry of
≤
,
{\displaystyle \,\leq ,}
a function can have at most one maximal value.
^ This is an identity between sets, more particularly, between subsets of
Y
.
{\displaystyle Y.}
^ Note that
x
(
10
−
x
)
=
25
−
(
x
−
5
)
2
≤
25
{\displaystyle x(10-x)=25-(x-5)^{2}\leq 25}
with equality if and only if
x
−
5
=
0.
{\displaystyle x-5=0.}
References
[ edit ]
External links
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Arg_max&oldid=1225875746 "
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