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:
- Strong and weak laws of large numbers for Fréchet sample means in bounded metric spaces
- Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables
- Averaging metric phylogenetic trees
- Central limit theorems for Fréchet means in the space of phylogenetic trees
- A stochastic algorithm finding \( {p} \)-means on the circle
- Means in complete manifolds: uniqueness and approximation
- Consistent estimation of a population barycenter in the Wasserstein space
- Minimax properties of Fréchet means of discretely sampled curves
- Fréchet means of curves for signal averaging and application to ECG data analysis
- On the consistency of Fréchet means in deformable models for curve and image analysis
- Distribution's template estimate with Wasserstein metrics