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Let $X$ be a set of $n$ points in $\mathbb{R}^3$ and $f_m$ be the Fréchet mean, i.e.:

$$ f_m= \arg \min_{p \in M} \sum_{i=1}^n w_id^2(p,x_i) $$

where $(\mathbb{R}^3,d)$ is a complete metric space, $d$ a distance function (Euclidean $d_E$, log-Euclidean $d_{log}$, Riemannian $d_R$) between any two points and $w$ a weight function.

$$ d_E(x_i,x_j)= \| x_i - x_j\| $$

$$ d_{log}(x_i,x_j)= \| \log(x_i) - \log(x_j)\| $$

$$ d_R(x_i,x_j)= \| \log(x_i^{-1/2}x_jx_i^{-1/2})\| $$

How could I solve this problem? Maybe by Gradient Descent?

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    If the points are in $\mathbb R^3$, what does $(M,d)$ have to do with the problem?2012-06-05
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    You are right. It is $(\mathbb{R}^3,d)$2012-06-05
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    OK, is $d(x,y)$ the Euclidean distance $\sqrt{(x_1-y_1)^2+\dots+(x_n-y_n)^2}$ or someone else?2012-06-05
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    I was vague bacause I would like to analize the differences between Euclidean, log-Euclidean, and Riemannian metric.2012-06-05

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