# Fréchet mean of the hyperbolic shape space

The Fréchet mean of a general subspace is defined as $$F(x)=\int_Mdist(x,y)^2d\mu(y),$$ where $\mu$ is the probability measure on a general metric space $(M,dist)$.

I understand that the Fréchet mean for the hyperbolic space is defined as $$\int\cosh(\sqrt{2}\delta([X],[Y]))d\mu([Y]),$$ where $\delta$ denotes the induced Riemannian metric of the hyperbolic space.

However, I have also come across a Fréchet mean (for the hyperbolic space) defined as $$\int\sinh(\delta([X],[Y])/\sqrt{2})^2d\mu([Y]).$$

What is the difference between these two means and how are they obtained?

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If we have a sample, say $(y_1, ..., y_i)$, then the sample mean can be defined by $$\hat{F}=\sum_{i=1}^nd^2(x,y_i).$$ I think we need to substitute hyperbolic coordinates $((x_1, y_1), ..., (x_i, y_i))$ into the above function and then by manipulating it form the simpler function of $$\int\cosh(\sqrt{2}\delta([X],[Y]))d\mu([Y]).$$ Not entirely sure though, any thoughts on this will be much appreciated. –  Luis_G Jan 31 at 22:32