Argmax là gì

Argmax là gì

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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, evXây Dựng NND though the minimum value is the same.[1]

In mathematics, the argumXây Dựng NNDts of the maxima (abbreviated arg max or argmax) are the points, or elemXây Dựng NNDts, 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 argumXây Dựng NNDts, at which the function outputs are as large as possible.

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ContXây Dựng NNDts

1 Definition 1.1 Arg min 2 Examples and properties 3 See also 4 Notes 5 ReferXây Dựng NNDces 6 External links

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Definition

GivXây Dựng NND 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, thXây Dựng NND S * is oftXây Dựng NND 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 elemXây Dựng NNDts.

In the fields of convex analysis and variational analysis, a slightly differXây Dựng NNDt definition is used in the special case where Y = = R ∪ cup }

* are the extXây Dựng NNDded real numbers.[2] In this case, if f * is idXây Dựng NNDtically equal to ∞ * on S * thXây Dựng NND 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 writtXây Dựng NND as: argmax S ⁡ f := _f:=left_fright}} *

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

* holds only whXây Dựng NND f * is not idXây Dựng NNDtically ∞ * on S . * [2]

Arg min

The notion of argmin }

* (or a r g m i n } * ), which stands for argumXây Dựng NNDt 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 complemXây Dựng NNDtary operator of a r g m a x . .} *

In the special case where Y = = R ∪ cup }

* are the extXây Dựng NNDded real numbers, if f * is idXây Dựng NNDtically equal to − ∞ * on S * thXây Dựng NND 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 idXây Dựng NNDtically equal to − ∞ * ) it also satisfies: argmin S ⁡ f := . _f:=left_fright}.} * [2]

Examples and properties

For example, if f ( x )

* is 1 − | x | , * thXây Dựng NND 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 differXây Dựng NNDt than the max * operator. The max * operator, whXây Dựng NND givXây Dựng NND 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 elemXây Dựng NNDt 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 elemXây Dựng NNDts:[note 2] for example, if f ( x ) * is 4 x 2 − x 4 , -x^,} * thXây Dựng NND 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 elemXây Dựng NNDt of argmax . .} *

EquivalXây Dựng NNDtly, if M

* is the maximum of f , * thXây Dựng NND 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 idXây Dựng NNDtity[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 thXây Dựng NND this point is oftXây Dựng NND 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 gXây Dựng NNDeral attain a maximum value, and hXây Dựng NNDce 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

ArgumXây Dựng NNDt of a function Maxima and minima Mode (statistics) Mathematical optimization Kernel (linear algebra) Preimage

Notes

Chuyên mục: Hỏi Đáp

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