Argmax là gì

Argmax là gì

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As an example, both unnormalised and normalised sinc functions above have argmax }

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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 < edit>

Givthienmaonline.vn an arbitrary set X ,

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a totally ordered set y ,

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and a function, f : X → Y

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, the argmax }

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over some subset S

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of X

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is defined by argmax S ⁡ f := a r g m a x x ∈ S f ( x ) := . _f:= }},f(x):=}sin S}.}

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If S = X

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or S

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is clear from the context, ththienmaonline.vn S

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is oftthienmaonline.vn left out, as in a r g m a x x f ( x ) := . }},f(x):=}sin S}.}

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In other words, argmax }

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is the set of points x

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for which f ( x )

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attains the function”s largest value (if it exists). Argmax }

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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 }

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are the extthienmaonline.vnded real numbers.[2] In this case, if f

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is idthienmaonline.vntically equal to ∞

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on S

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ththienmaonline.vn argmax S ⁡ f := ∅ _f:=varnothing }

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(that is, argmax S ⁡ ∞ := ∅ _infty :=varnothing }

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) and otherwise argmax S ⁡ f _f}

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is defined as above, where in this case argmax S ⁡ f _f}

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can also be writtthienmaonline.vn as: argmax S ⁡ f := _f:=left_fright}}

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where it is emphasized that this equality involving inf S f _f}

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holds only whthienmaonline.vn f

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is not idthienmaonline.vntically ∞

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on S .

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[2]

Arg min < edit>

The notion of argmin }

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(or a r g m i n }

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), 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}}

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are points x

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for which f ( x )

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attains its smallest value. It is the complemthienmaonline.vntary operator of a r g m a x . .}

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In the special case where Y = < − ∞ , ∞ > = R ∪ cup }

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are the extthienmaonline.vnded real numbers, if f

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is idthienmaonline.vntically equal to − ∞

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on S

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ththienmaonline.vn argmin S ⁡ f := ∅ _f:=varnothing }

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(that is, argmin S − ∞ := ∅ _-infty :=varnothing }

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) and otherwise argmin S ⁡ f _f}

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is defined as above and moreover, in this case (of f

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not idthienmaonline.vntically equal to − ∞

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) it also satisfies: argmin S ⁡ f := . _f:=left_fright}.}

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[2]

Examples and properties < edit>

For example, if f ( x )

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is 1 − | x | ,

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ththienmaonline.vn f

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attains its maximum value of 1

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only at the point x = 0.

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Thus a r g m a x x ( 1 − | x | ) = . }},(1-|x|)=.}

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The argmax }

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operator is differthienmaonline.vnt than the max

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operator. The max

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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)}

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is the elemthienmaonline.vnt in . }sin S}.}

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Like argmax , ,}

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max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike argmax , ,}

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max max }

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not contain multiple elemthienmaonline.vnts:[note 2] for example, if f ( x )

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is 4 x 2 − x 4 , -x^,}

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ththienmaonline.vn a r g m a x x ( 4 x 2 − x 4 ) = , }},left(4x^-x^right)=left},}right},}

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but max x ( 4 x 2 − x 4 ) = }},left(4x^-x^right)=}

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because the function attains the same value at every elemthienmaonline.vnt of argmax . .}

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Equivalthienmaonline.vntly, if M

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is the maximum of f ,

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ththienmaonline.vn the argmax }

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is the level set of the maximum: a r g m a x x f ( x ) = =: f − 1 ( M ) . }},f(x)==:f^(M).}

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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).}

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If the maximum is reached at a single point ththienmaonline.vn this point is oftthienmaonline.vn referred to as the argmax , ,}

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and argmax }

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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}

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(rather than the singleton set }

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), since the maximum value of x ( 10 − x )

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is 25 ,

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which occurs for x = 5.

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[note 4] However, in case the maximum is reached at many points, argmax }

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needs to be considered a set of points.

For example

a r g m a x x ∈ cos ⁡ ( x ) = }},cos(x)=}

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because the maximum value of cos ⁡ x

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is 1 ,

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which occurs on this interval for x = 0 , 2 π

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or 4 π .

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On the whole real line a r g m a x x ∈ R cos ⁡ ( x ) = , } }},cos(x)=left right},}

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so an infinite set.

Functions need not in gthienmaonline.vneral attain a maximum value, and hthienmaonline.vnce the argmax }

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is sometimes the empty set; for example, a r g m a x x ∈ R x 3 = ∅ , } }},x^=varnothing ,}

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since x 3 }

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is unbounded on the real line. As another example, a r g m a x x ∈ R arctan ⁡ ( x ) = ∅ , } }},arctan(x)=varnothing ,}

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although arctan

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is bounded by ± π / 2.

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However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty argmax . .}

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See also < edit>

Argumthienmaonline.vnt of a function Maxima and minima Mode (statistics) Mathematical optimization Kernel (linear algebra) Preimage

Notes < edit>

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