Geometric median

For another use of the term median in geometry, see Median (geometry).

The geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions. It is also known as the Fermat–Weber point or 1-median.

The geometric median is an important estimator of location in statistics. It is also a standard problem in facility location, where it models the problem of locating a facility to minimize the cost of transportation.

The special case of the problem for three points in the plane (that is, m = 3 and n = 2) is sometimes also known as Fermat's problem; it arises in the construction of minimal Steiner trees, and was originally posed as a problem by Pierre de Fermat to Evangelista Torricelli, who solved it. Its solution is now known as the Fermat point of the triangle formed by the three sample points. Alfred Weber's name is associated with the more general Fermat–Weber problem due to a discussion of the problem in his 1909 book on facility location.

Wesolowsky (1993) provides a survey of the problem. See Fekete, Mitchell, and Beurer (2003) for generalizations of the problem to non-discrete point sets.

Definition

Formally, for a given set of m points $x_{1},x_{2},\dots ,x_{m}\,$ with each $x_{i}\in \mathbb {R} ^{n}$, the geometric median is defined as

Geometric Median $={\underset {y\in \mathbb {R} ^{n}}{\operatorname {arg\,min} }}\sum _{i=1}^{m}\left\|x_{i}-y\right\|_{2}$

Note that argmin means the argument for which the sum is minimized. In this case, it is the point $y$ from where the sum of all Euclidean distances to the $x_{i}$'s is minimum.

Properties

• For the 1-dimensional case, the geometric median coincides with the median. This is because the univariate median also minimizes the sum of distances from the points.
• The geometric median is unique whenever the points are not collinear.
• The geometric median is equivariant for Euclidean similarity transformations, including translation and rotation. This means that one would get the same result either by transforming the geometric median, or by applying the same transformation to the sample data and finding the geometric median of the transformed data. This property follows from the fact that the geometric median is defined only from pairwise distances, and doesn't depend on the system of orthogonal Cartesian coordinates by which the sample data is represented. In contrast, the component-wise median for a multivariate data set is not in general rotation invariant, nor is it independent of the choice of coordinates.
• The geometric median has a breakdown point of 0.5. That is, up to half of the sample data may be arbitrarily corrupted, and the median of the samples will still provide a robust estimator for the location of the uncorrupted data.

Special cases

• For 3 points, if any angle of the triangle is more than 120° then the geometric median is the point making that angle. If all the angles are less than 120°, the geometric median is the point inside the triangle which subtends an angle of 120° to all three pairs of points. This is also known as the Fermat point of the triangle formed by the three points.
• For 4 coplanar points, if one of the four points is inside the triangle formed by the other three points, then the geometric median is that point. Otherwise, the points form a convex quadrilateral and the geometric median is the crossing point of the diagonals of the quadrilateral. The geometric median of four coplanar points is the same as the unique Radon point of the four points.

Computation

Despite being an easy to understand concept, computing the geometric median poses a challenge. The centroid or center of mass, defined similarly to the geometric median as minimizing the sum of the squares of the distances to each sample, can be found by a simple formula — its coordinates are the averages of the coordinates of the samples — but no such formula is known for the geometric median, and it has been shown that no formula involving only arithmetic operations and kth roots can exist in general.

However, it is straightforward to calculate an approximation to the geometric median using an iterative procedure in which each step produces a more accurate approximation. Procedures of this type can be derived from the fact that the sum of distances is a convex function, since the distance to each sample point is convex and the sum of convex functions remains convex. Therefore, procedures that decrease the sum of distances at each step cannot get trapped in a local optimum.

One common approach of this type, called Weiszfeld's algorithm, is a form of iteratively re-weighted least squares. This algorithm defines a set of weights that are inversely proportional to the distances from the current estimate to the samples, and creates a new estimate that is the weighted average of the samples according to these weights. That is,

$\left.y_{i+1}=\left(\sum _{j=1}^{m}{\frac {x_{j}}{\|x_{j}-y_{i}\|}}\right)\right/\left(\sum _{j=1}^{m}{\frac {1}{\|x_{j}-y_{i}\|}}\right).$

Bose et al. (2003) describe more sophisticated geometric optimization procedures for finding approximately optimal solutions to this problem. As Nie, Parrilo & Sturmfels (2008) show, the problem can also be represented as a semidefinite program.

Implicit formula

If y is distinct from all the given points, x_j, then y is the geometric median if and only if it satisfies:

$0=\sum _{j=1}^{m}{\frac {x_{j}-y}{\left\|x_{j}-y\right\|}}.$

This is equivalent to:

$\left.y=\left(\sum _{j=1}^{m}{\frac {x_{j}}{\|x_{j}-y\|}}\right)\right/\left(\sum _{j=1}^{m}{\frac {1}{\|x_{j}-y\|}}\right)$

which is closely related to Weiszfeld's algorithm.

If y is equal to some of the given points, then y is the geometric median if and only if there are vectors u_j such that:

$0=\sum _{j=1}^{m}u_{j}$

where for x_j ≠ y,

$u_{j}={\frac {x_{j}-y}{\left\|x_{j}-y\right\|}}$

and for x_j = y,

$\|u_{j}\|\leq 1.$

Generalizations

The geometric median can be generalized from Euclidean spaces to general Riemannian manifolds (and even metric spaces) using the same idea which is used to define the Fréchet mean on a Riemannian manifold. Let $M$ be a Riemannian manifold with corresponding distance function $d(\cdot ,\cdot )$, let $w_{1},\ldots ,w_{n}$ be $n$ weights summing to 1, and let $x_{1},\ldots ,x_{n}$ be $n$ observations from $M$. Then we define the weighted geometric median $m$ (or weighted Fréchet median) of the data points as

$m={\underset {x\in M}{\operatorname {arg\,min} }}\sum _{i=1}^{n}w_{i}d(x,x_{i})$.

If all the weights are equal, we say simple that $m$ is the geometric median.

See also

• Central tendency
• Centroid, which minimizes the sum of squares of Euclidean distance
• mean
• Fréchet mean

References

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• Bajaj, C. (1988). "The algebraic degree of geometric optimization problems". Discrete and Computational Geometry. 3: 177–191. doi:10.1007/BF02187906.

• Bose, Prosenjit; Maheshwari, Anil; Morin, Pat (2003). "Fast approximations for sums of distances, clustering and the Fermat–Weber problem". Computational Geometry: Theory and Applications. 24 (3): 135–146. doi:10.1016/S0925-7721(02)00102-5.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Chandrasekaran, R.; Tamir, A. (1989). "Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem". Mathematical Programming, Series A. 44: 293–295. doi:10.1007/BF01587094.{{cite journal}}: CS1 maint: multiple names: authors list (link)

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• Fekete, Sándor P.; Mitchell, Joseph S. B.; Beurer, Karin (2003). "On the continuous Fermat-Weber problem". arXiv:cs.CG/0310027. {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: multiple names: authors list (link)

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• Lopuhaä, Hendrick P.; Rousseeuw, Peter J. (1991). "Breakdown points of affine equivariant estimators of multivariate location and covariance matrices". Annals of Statistics. 19 (1): 229–248. doi:10.1214/aos/1176347978.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Nie, Jiawang; Parrilo, Pablo A.; Sturmfels, Bernd (2008), "Semidefinite representation of the k-ellipse", in Dickenstein, A.; Schreyer, F.-O.; Sommese, A.J. (eds.), Algorithms in Algebraic Geometry, IMA Volumes in Mathematics and its Applications, vol. 146, Springer-Verlag, pp. 117–132, arXiv:math/0702005.

• Ostresh, L. (1978). "Convergence of a Class of Iterative Methods for Solving Weber Location Problem". Operations Research. 26: 597–609.

• Weber, Alfred (1909). Über den Standort der Industrien, Erster Teil: Reine Theorie des Standortes. Tübingen: Mohr.

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• Weiszfeld, E. (1937). "Sur le point pour lequel la somme des distances de n points donnes est minimum". Tohoku Math. Journal. 43: 355–386.

• P Thomas Fletcher, Suresh Venkatasubramanian, Sarang Joshi (2009). "The geometric median on Riemannian manifolds with applications to robust atlas estimation". Neuroimage. 45: s143–s152.{{cite journal}}: CS1 maint: multiple names: authors list (link)