Fréchet mean

In mathematics and statistics, the Fréchet mean is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. Karcher means are a closely related construction named after Hermann Karcher.^[1] On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.

Definition

Let (M, d) be a complete metric space. Let x₁, x₂, ..., x_N be random 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:

$\Psi(p) = \sum_{i=1}^N d^2(p, x_i)$

Define the Fréchet mean to be the point m of M at which the variance Ψ is minimized, if this minimizer is unique:

$m = \mathop{\mathrm{arg\ min}}_{p\in M} \sum_{i=1}^N d^2(p, x_i)$

A Karcher mean is a local minimum of the Fréchet variance.^[1] Sometimes the x_i are assigned weights w_i. In that case,

$\Psi(p) = \sum_{i=1}^N w_i d^2(p, x_i), \;\;\;\; m = \mathop{\mathrm{arg\ min}}_{p\in M} \sum_{i=1}^N w_i d^2(p, x_i).$

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, using the square root of the distance.^[2]

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) $d_H(x,y) = \left| \frac{1}{x} - \frac{1}{y} \right|$ 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

$d_m(x,y)=| x^m - y^m |$.

f-mean

Given an invertible function $f$, the f-mean can be defined as the Fréchet mean obtained by using the metric $d_f(x,y)= | f(x)-f(y)|$. This is sometimes called the Generalised f-mean or Quasi-arithmetic mean.

Weighted means

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

References

1. Nielsen, Frank; Bhatia, Rajendra (2012), Matrix Information Geometry, Springer, p. 171, ISBN 9783642302329.
2. Nielsen & Bhatia (2012), p. 136.