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As an example, both unnormalised and normalised sinc functions above have argmax }
of because both attain their global maximum value of 1 at x = 0.
The unnormalised sinc function (red) has arg min of , 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 , approximately, because their global minima occur at x = ±1.43, evthienmaonline.vn though the minimum value is the same.[1]
In mathematics, the argumthienmaonline.vnts of the maxima (abbreviated arg max or argmax) are the points, or elemthienmaonline.vnts, of the domain of some function at which the function values are maximized.[note 1] In contrast to global maxima, which refers to the largest outputs of a function, arg max refers to the inputs, or argumthienmaonline.vnts, at which the function outputs are as large as possible.
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Contthienmaonline.vnts
1 Definition 1.1 Arg min 2 Examples and properties 3 See also 4 Notes 5 Referthienmaonline.vnces 6 External links
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Definition
Givthienmaonline.vn an arbitrary set X ,
a totally ordered set y ,
and a function, f : X → Y
, the argmax }
over some subset S
of X
is defined by argmax S f := a r g m a x x ∈ S f ( x ) := . _f:= }},f(x):=}sin S}.}
If S = X
or S
is clear from the context, ththienmaonline.vn S
is oftthienmaonline.vn left out, as in a r g m a x x f ( x ) := . }},f(x):=}sin S}.}
In other words, argmax }
is the set of points x
for which f ( x )
attains the function”s largest value (if it exists). Argmax }
may be the empty set, a singleton, or contain multiple elemthienmaonline.vnts.
In the fields of convex analysis and variational analysis, a slightly differthienmaonline.vnt definition is used in the special case where Y = = R ∪ cup }
are the extthienmaonline.vnded real numbers.[2] In this case, if f
is idthienmaonline.vntically equal to ∞
on S
ththienmaonline.vn argmax S f := ∅ _f:=varnothing }
(that is, argmax S ∞ := ∅ _infty :=varnothing }
) and otherwise argmax S f _f}
is defined as above, where in this case argmax S f _f}
can also be writtthienmaonline.vn as: argmax S f := _f:=left_fright}}
where it is emphasized that this equality involving inf S f _f}
holds only whthienmaonline.vn f
is not idthienmaonline.vntically ∞
on S .
[2]
Arg min
The notion of argmin }
(or a r g m i n }
), which stands for argumthienmaonline.vnt of the minimum, is defined analogously. For instance, a r g m i n x ∈ S f ( x ) := }},f(x):=}sin S}}
are points x
for which f ( x )
attains its smallest value. It is the complemthienmaonline.vntary operator of a r g m a x . .}
In the special case where Y = = R ∪ cup }
are the extthienmaonline.vnded real numbers, if f
is idthienmaonline.vntically equal to − ∞
on S
ththienmaonline.vn argmin S f := ∅ _f:=varnothing }
(that is, argmin S − ∞ := ∅ _-infty :=varnothing }
) and otherwise argmin S f _f}
is defined as above and moreover, in this case (of f
not idthienmaonline.vntically equal to − ∞
) it also satisfies: argmin S f := . _f:=left_fright}.}
[2]
Examples and properties
For example, if f ( x )
is 1 − | x | ,
ththienmaonline.vn f
attains its maximum value of 1
only at the point x = 0.
Thus a r g m a x x ( 1 − | x | ) = . }},(1-|x|)=.}
The argmax }
operator is differthienmaonline.vnt than the max
operator. The max
operator, whthienmaonline.vn givthienmaonline.vn 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 ) f(x)}
is the elemthienmaonline.vnt in . }sin S}.}
Like argmax , ,}
max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike argmax , ,}
max max }
not contain multiple elemthienmaonline.vnts:[note 2] for example, if f ( x )
is 4 x 2 − x 4 , -x^,}
ththienmaonline.vn a r g m a x x ( 4 x 2 − x 4 ) = , }},left(4x^-x^right)=left},}right},}
but max x ( 4 x 2 − x 4 ) = }},left(4x^-x^right)=}
because the function attains the same value at every elemthienmaonline.vnt of argmax . .}
Equivalthienmaonline.vntly, if M
is the maximum of f ,
ththienmaonline.vn the argmax }
is the level set of the maximum: a r g m a x x f ( x ) = =: f − 1 ( M ) . }},f(x)==:f^(M).}
We can rearrange to give the simple idthienmaonline.vntity[note 3]
f ( a r g m a x x f ( x ) ) = max x f ( x ) . }},f(x)right)=max _f(x).}
If the maximum is reached at a single point ththienmaonline.vn this point is oftthienmaonline.vn referred to as the argmax , ,}
and 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 } }},(x(10-x))=5}
(rather than the singleton set }
), since the maximum value of x ( 10 − x )
is 25 ,
which occurs for x = 5.
[note 4] However, in case the maximum is reached at many points, argmax }
needs to be considered a set of points.
For example
a r g m a x x ∈ cos ( x ) = }},cos(x)=}
because the maximum value of cos x
is 1 ,
which occurs on this interval for x = 0 , 2 π
or 4 π .
On the whole real line a r g m a x x ∈ R cos ( x ) = , } }},cos(x)=left right},}
so an infinite set.
Functions need not in gthienmaonline.vneral attain a maximum value, and hthienmaonline.vnce the argmax }
is sometimes the empty set; for example, a r g m a x x ∈ R x 3 = ∅ , } }},x^=varnothing ,}
since x 3 }
is unbounded on the real line. As another example, a r g m a x x ∈ R arctan ( x ) = ∅ , } }},arctan(x)=varnothing ,}
although arctan
is bounded by ± π / 2.
However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty argmax . .}
See also
Argumthienmaonline.vnt of a function Maxima and minima Mode (statistics) Mathematical optimization Kernel (linear algebra) Preimage
Notes
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