Английская Википедия:Arg max
Шаблон:Short description Шаблон:Refimprove
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
Given an arbitrary set Шаблон:Nowrap a totally ordered set Шаблон:Nowrap and a function, Шаблон:Nowrap the <math>\operatorname{argmax}</math> over some subset <math>S</math> of <math>X</math> is defined by
- <math>\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 \}.</math>
If <math>S = X</math> or <math>S</math> is clear from the context, then <math>S</math> is often left out, as in <math>\underset{x}{\operatorname{arg\,max}}\, f(x) := \{ x ~:~ f(s) \leq f(x) \text{ for all } s \in X \}.</math> In other words, <math>\operatorname{argmax}</math> is the set of points <math>x</math> for which <math>f(x)</math> attains the function's largest value (if it exists). <math>\operatorname{Argmax}</math> 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 <math>Y = [-\infty,\infty] = \mathbb{R} \cup \{ \pm\infty \}</math> are the extended real numbers.Шаблон:Sfn In this case, if <math>f</math> is identically equal to <math>\infty</math> on <math>S</math> then <math>\operatorname{argmax}_S f := \varnothing</math> (that is, <math>\operatorname{argmax}_S \infty := \varnothing</math>) and otherwise <math>\operatorname{argmax}_S f</math> is defined as above, where in this case <math>\operatorname{argmax}_S f</math> can also be written as:
- <math>\operatorname{argmax}_S f := \left\{ x \in S ~:~ f(x) = \sup {}_S f \right\}</math>
where it is emphasized that this equality involving <math>\sup {}_S f</math> holds Шаблон:Em when <math>f</math> is not identically <math>\infty</math> on Шаблон:NowrapШаблон:Sfn
Arg min
The notion of <math>\operatorname{argmin}</math> (or <math>\operatorname{arg\,min}</math>), which stands for argument of the minimum, is defined analogously. For instance,
- <math>\underset{x \in S}{\operatorname{arg\,min}} \, f(x) := \{ x \in S ~:~ f(s) \geq f(x) \text{ for all } s \in S \}</math>
are points <math>x</math> for which <math>f(x)</math> attains its smallest value. It is the complementary operator of Шаблон:Nowrap
In the special case where <math>Y = [-\infty,\infty] = \R \cup \{ \pm\infty \}</math> are the extended real numbers, if <math>f</math> is identically equal to <math>-\infty</math> on <math>S</math> then <math>\operatorname{argmin}_S f := \varnothing</math> (that is, <math>\operatorname{argmin}_S -\infty := \varnothing</math>) and otherwise <math>\operatorname{argmin}_S f</math> is defined as above and moreover, in this case (of <math>f</math> not identically equal to <math>-\infty</math>) it also satisfies:
- <math>\operatorname{argmin}_S f := \left\{ x \in S ~:~ f(x) = \inf {}_S f \right\}.</math>Шаблон:Sfn
Examples and properties
For example, if <math>f(x)</math> is <math>1 - |x|,</math> then <math>f</math> attains its maximum value of <math>1</math> only at the point <math>x = 0.</math> Thus
- <math>\underset{x}{\operatorname{arg\,max}}\, (1 - |x|) = \{ 0 \}.</math>
The <math>\operatorname{argmax}</math> operator is different from the <math>\max</math> operator. The <math>\max</math> operator, when given the same function, returns the Шаблон:Em of the function instead of the Шаблон:Em that cause that function to reach that value; in other words
- <math>\max_x f(x)</math> is the element in <math>\{ f(x) ~:~ f(s) \leq f(x) \text{ for all } s \in S \}.</math>
Like <math>\operatorname{argmax},</math> max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike <math>\operatorname{argmax},</math> <math>\operatorname{max}</math> may not contain multiple elements:[note 2] for example, if <math>f(x)</math> is <math>4 x^2 - x^4,</math> then <math>\underset{x}{\operatorname{arg\,max}}\, \left( 4 x^2 - x^4 \right) = \left\{-\sqrt{2}, \sqrt{2}\right\},</math> but <math>\underset{x}{\operatorname{max}}\, \left( 4 x^2 - x^4 \right) = \{ 4 \}</math> because the function attains the same value at every element of <math>\operatorname{argmax}.</math>
Equivalently, if <math>M</math> is the maximum of <math>f,</math> then the <math>\operatorname{argmax}</math> is the level set of the maximum:
- <math>\underset{x}{\operatorname{arg\,max}} \, f(x) = \{ x ~:~ f(x) = M \} =: f^{-1}(M).</math>
We can rearrange to give the simple identity[note 3]
- <math>f\left(\underset{x}{\operatorname{arg\,max}} \, f(x) \right) = \max_x f(x).</math>
If the maximum is reached at a single point then this point is often referred to as Шаблон:Em <math>\operatorname{argmax},</math> and <math>\operatorname{argmax}</math> is considered a point, not a set of points. So, for example,
- <math>\underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, (x(10 - x)) = 5</math>
(rather than the singleton set <math>\{ 5 \}</math>), since the maximum value of <math>x (10 - x)</math> is <math>25,</math> which occurs for <math>x = 5.</math>[note 4] However, in case the maximum is reached at many points, <math>\operatorname{argmax}</math> needs to be considered a Шаблон:Em of points.
For example
- <math>\underset{x \in [0, 4 \pi]}{\operatorname{arg\,max}}\, \cos(x) = \{ 0, 2 \pi, 4 \pi \}</math>
because the maximum value of <math>\cos x</math> is <math>1,</math> which occurs on this interval for <math>x = 0, 2 \pi</math> or <math>4 \pi.</math> On the whole real line
- <math>\underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \cos(x) = \left\{ 2 k \pi ~:~ k \in \mathbb{Z} \right\},</math> so an infinite set.
Functions need not in general attain a maximum value, and hence the <math>\operatorname{argmax}</math> is sometimes the empty set; for example, <math>\underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, x^3 = \varnothing,</math> since <math>x^3</math> is unbounded on the real line. As another example, <math>\underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \arctan(x) = \varnothing,</math> although <math>\arctan</math> is bounded by <math>\pm\pi/2.</math> However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty <math>\operatorname{argmax}.</math>
See also
- Argument of a function
- Maxima and minima
- Mode (statistics)
- Mathematical optimization
- Kernel (linear algebra)
- Preimage
Notes
References
External links
- ↑ "The Unnormalized Sinc Function Шаблон:Webarchive", University of Sydney
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