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Mean of a random variable on a metric space

In this short post, we recall the pleasant notion of Fréchet mean (or Karcher mean) of a probability measure on a metric space, a concept already considered in an old previous post. Let \( {(E,d)} \) be a metric space, such as a graph (with vertices and edges) or a Riemannian manifold, equipped with its Borel \( {\sigma} \)-field. Let \( {\mu} \) be a probability measure on \( {E} \). How can we define the mean \( {m_\mu} \) and the variance \( {v_\mu} \) of \( {\mu} \)? A very natural idea is to consider the variational definition:

\[ v_\mu:=\inf_{x\in E}\mathbb{E}(d(x,Y)^2), \]

where \( {Y\sim\mu} \). The set \( {m_\mu:=\arg\inf_{x\in E}\mathbb{E}(d(x,Y)^2)} \) where this infimum is achieved plays the role of a mean (which is not necessarily unique), while the value of the infimum plays the role of the variance. Note that

\[v_\mu=\inf_{x\in E}W_2(\delta_x,\mu)^2\]

where $W_2$ is the so-called Wasserstein-Fréchet-Kantorovich coupling distance

\[ W_2(\nu,\mu)^2=\inf_{(X,Y),X\sim\nu,Y\sim\mu}\mathbb{E}(d(X,Y)^2).\]

From this observation, the Fréchet means of $\mu$ are the atoms of the closest Dirac masses to $\mu$ in $W_2$ distance. We may replace the exponent \( {2} \) by any real \( {p\geq1} \) to get a more general notion of moments of \( {\mu} \) (this leads by the way to moments problems on metric spaces!). The notion of Fréchet mean is so natural that it has been studied by various authors in pure and applied contexts. Here are some recent examples:

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