# Fréchet mean for a general shape space

I am posting this question in order to gain a better understand of what the Fréchet mean is for a generalised shape space.

So firstly I gather that the Fréchet mean of a probabilty measure $\mu$ on a general metric space $(M,dist)$ is a generalisation of the mean or the expectation of a probability distribution on a Euclidean space and is defined as any global minimum of the function $$F(x)=\int_Mdist(x,y)^2d\mu(y).$$ This concept can be further generalised by replacing 'dist' in the integrand by a suitable function of the distance.

My question is, what exactly does this formula do? I think it has something to do with minimising the mean between a data set, however I am not sure of this. Also I am not sure how I would use this formula. What exactly is the 'probability measure' in this case? I think it's some kind of statistical distribution, but again I am really not sure.

Any help with this question will be much appreiciated, thank you.

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If $x^*$ is an argument of a global minimum of $F$, then we know that the $\mu$-average distance from $x^*$ to any point $y\in M$ is minimal. – Ilya Jan 26 '13 at 21:45
Thank you, so $x^*$ is the argmin of F, would we consider $y$ to be the argmin of another data set? And what exactly is the $\mu$-average? I.e. what is a probability measure? – Luis_G Jan 26 '13 at 21:55
In order to have any concept of a mean we need a probability measure. When finding the mean of numbers $1,2,3,4$ we (implicitly) use the probability measure $\mu$ such that $\mu(\{k\})=1/4$ for $k=1,2,3,4$. So, if you need the mean of a finite subset of a metric space, using this soft of measure (normalized counting measure) may be appropriate. If the set is infinite, one has to think harder of what the appropriate measure should be. – user53153 Jan 27 '13 at 0:33