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I am doing the following exercise: given the level lines of the unimodal function $f$ with minimum $x^*$, a point $x_0$, and vectors $v_1$,$v_2$, $v_3$, $v_4$, $v_5$, one of which is equal to $\nabla f(x_0)$, which vector $v_i$ is equal to $\nabla f(x_0)$?

Exercise

My answer is $v_5$, because I know that the gradient of a function in a point $x_0$ must be perpendicular to level line in that point and the only two perpendicular vectors are $v_1$ and $v_5$. Furthermore, the gradient points towards the direction of maximum increase and $v_1$ is going towards the minimum $x^*$, therefore the correct answer should be $v_5$.

Am I correct or am I missing something?

cholo14
  • 451
  • I'd also say $v_5$ :). And you can see that gradient descent for example would follow a multiple times the negative of the gradient (so $\alpha v_1$) then recompute which does seem to go in the right direction (another way to check that you've got the right direction) – tibL Sep 16 '16 at 16:10

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