Fréchet mean

Definition

Let (M, d) be a complete metric space. Let x1, x2, …, x**N be points in M. For any point p in M, define the Fréchet variance to be the sum of squared distances from p to the x**i: :\Psi(p) = \sum_{i=1}^N d^2(p, x_i)

The Karcher means are then those points, m of M, which minimise Ψ: :m = \mathop{\text{arg min}}{p \in M} \sum{i=1}^N d^2(p, x_i)

If there is a unique m of M that strictly minimises Ψ, then it is Fréchet mean.

Examples of Fréchet means

Arithmetic mean and median

For real numbers, the arithmetic mean is a Fréchet mean, using the usual Euclidean distance as the distance function.

The median is also a Fréchet mean, if the definition of the function Ψ is generalized to the non-quadratic :\Psi(p) = \sum_{i=1}^N d^\alpha(p, x_i), where \alpha=1, and the Euclidean distance is the distance function d. In higher-dimensional spaces, this becomes the geometric median.

Geometric mean

On the positive real numbers, the (hyperbolic) distance function d(x,y)= | \log(x) - \log(y) | can be defined. The geometric mean is the corresponding Fréchet mean. Indeed f:x\mapsto e^x is then an isometry from the euclidean space to this "hyperbolic" space and must respect the Fréchet mean: the Fréchet mean of the x_i is the image by f of the Fréchet mean (in the Euclidean sense) of the f^{-1}(x_i), i.e. it must be:

: f\left( \frac{1}{n}\sum_{i=1}^n f^{-1}(x_i)\right) = \exp \left( \frac{1}{n} \sum_{i=1}^n\log x_i \right) = \sqrt[n]{x_1 \cdots x_n}.

Harmonic mean

On the positive real numbers, the metric (distance function): : d_\operatorname{H}(x,y) = \left| \frac{1}{x} - \frac{1}{y} \right|

can be defined. The harmonic mean is the corresponding Fréchet mean.

Power means

Given a non-zero real number m, the power mean can be obtained as a Fréchet mean by introducing the metric : d_m(x, y) = \left| x^m - y^m \right|

f-mean

Given an invertible and continuous function f, the f-mean can be defined as the Fréchet mean obtained by using the metric: : d_f(x,y) = \left|f(x) - f(y)\right|

This is sometimes called the generalised f-mean or quasi-arithmetic mean.

Weighted means

The general definition of the Fréchet mean (which includes the possibility of weighting observations) can be used to derive weighted versions for all of the above types of means.

For the arithmetic mean, the x**i are assigned weights w**i. Then, the Fréchet variances and the Fréchet mean are defined as: :\Psi(p) = \sum_{i=1}^N w_i, d^2(p, x_i), ;;;; m = \mathop{\text{arg min}}{p \in M} \sum{i=1}^N w_i d^2(p, x_i).

References

References

1. (1973). "How to conjugate C1-close group actions, Math.Z. 132". Mathematische Zeitschrift.
2. (2012). "Matrix Information Geometry". Springer.
3. {{harvtxt. Nielsen. Bhatia