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My question is about two different definitions Gateaux derivative. I have seen the following two definitions but whether they are equivalent or which one is better to use I am not sure about:

Definition 1: Derivative at $f$ with increment $h$ is defined as \begin{align*} \lim_{ \theta \to 0} \frac{F[f+\theta h]- F[f]}{\theta} \end{align*}

Definition 2: Derivative at $f$ with increment $h$ is defined as \begin{align*} \lim_{ \theta \to 0} \frac{F[(1-\theta)f+\theta h]- F[f]}{\theta} \end{align*}

Are these the same? Is there an advantage of using Def 1 vs Def 2?

Def.1 is how I been thought the Gateaux derivative over vector space.

Def.2 I seen in the context of probability theory where derivative is taken over all probability distributions. The paper I read is also calling it Gateaux derivative.

I feel that the second one use if we want to take derivative over convex sets only. That is why we use $(1-\theta)f+\theta h$. But not sure completely?

Another question is if stationary point with one definition is also a stationary point with another?

Thank you in advance for any help.

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    $\begingroup$ If you replace $h$ by $h - f$ in the first definition, you arrive at the second. Hence, they define different things. The typical definition is the first one. $\endgroup$
    – gerw
    Mar 6, 2015 at 21:10
  • $\begingroup$ Thank you. Do you possibly know what is the second definition called? I saw it in a paper they referred to it as weak derivative. But didn't give any reference to the origin. $\endgroup$
    – Boby
    Mar 6, 2015 at 21:11
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    $\begingroup$ Derivative with increment $h-f$ ;) $\endgroup$
    – gerw
    Mar 6, 2015 at 21:12
  • $\begingroup$ @gerw Have you hear of weak derivative? $\endgroup$
    – Boby
    Mar 6, 2015 at 21:17
  • $\begingroup$ Yes, I know weak derivatives. $\endgroup$
    – gerw
    Mar 7, 2015 at 20:36

1 Answer 1

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The Gâteaux derivative has a standard definition, which is your definition 1. It may be the case that some author in a particular situation is envisaging the limit in your definition 2, which is certainly not far away from the "official" Gâteaux derivative. But there is an essential difference: The "official" derivative is linear in the variable $h$, while the expression in definition 2 is not.

We see the same discrepancy when considering Taylor expansions in ordinary calculus. The $n$-jet, i.e., the $n$-th degree Taylor polynomial, of a function $f$ at some point $a$ can be written as a polynomial in the "primary" variable $x$ as $$j_a^{n}(x)=\sum_{k=0}^n {f^{(k)}(a)\over k!}(x-a)^k\ ,$$ but equivalently as a polynomial in the increment variable $X:=x-a$: $$j_a^{n}(X)=\sum_{k=0}^n {f^{(k)}(a)\over k!}X^k\ .$$ The second view, which correspnds to your definition 1, seems much more natural.

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  • $\begingroup$ Dear Chirstian thank you very much for your answer. Do you think that results in optimization like Lagrange or KKT conditions are the same for def 1 and def 2? Or today have to be some how modified for def 2? $\endgroup$
    – Boby
    Mar 17, 2015 at 13:29
  • $\begingroup$ I'd not dare to use definition 2 without really thinking about its meaning beforehand. $\endgroup$ Mar 17, 2015 at 14:02
  • $\begingroup$ So, I am trying to use Gateaux derivative over space of probability distributions. I was researching the literature and they never use Def.1 but instead use Def. 2. However, I was not able to find why or the original source that say why its ok to use Def.2 instead of Def. 1. $\endgroup$
    – Boby
    Mar 17, 2015 at 14:06

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