# Geometric median

**minimizing the sum of distances**

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.wikipedia

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### Median

**averagesample medianmedian-unbiased estimator**

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. 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.

A geometric median, on the other hand, is defined in any number of dimensions.

### Fermat point

**FermatFermat-Torricelli pointsTorricelli–Fermat point**

Its solution is now known as the Fermat point of the triangle formed by the three sample points.

The Fermat point gives a solution to the geometric median and Steiner tree problems for three points.

### Weber problem

**a least cost theorySimpson-Weber triangle problemWeber**

The geometric median may in turn be generalized to the problem of minimizing the sum of weighted distances, known as the Weber problem after Alfred Weber's discussion of the problem in his 1909 book on facility location.

The Weber problem generalizes the geometric median, which assumes transportation costs per unit distance are the same for all destination points, and the problem of computing the Fermat point, the geometric median of three points.

### Central tendency

**Localitycentral locationcentral point**

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.

Geometric median: which minimizes the sum of distances to the data points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions.

### Alfred Weber

**AlfredWeber**

The geometric median may in turn be generalized to the problem of minimizing the sum of weighted distances, known as the Weber problem after Alfred Weber's discussion of the problem in his 1909 book on facility location.

Geometric median (also known as the Fermat–Weber problem)

### Andrew Vázsonyi

**Vázsonyi, Andrew**

One common approach of this type, called Weiszfeld's algorithm after the work of Endre Weiszfeld, is a form of iteratively re-weighted least squares.

He is known for Weiszfeld's algorithm for minimizing the sum of distances to a set of points, and for founding The Institute of Management Sciences.

### Radon's theorem

**minimal Radon partitionsRadonRadon's partition theorem**

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 four 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.

The Radon point of any four points in the plane is their geometric median, the point that minimizes the sum of distances to the other points.

### Iteratively reweighted least squares

**iteratively re-weighted least squaresIRLS**

One common approach of this type, called Weiszfeld's algorithm after the work of Endre Weiszfeld, is a form of iteratively re-weighted least squares.

Although not a linear regression problem, Weiszfeld's algorithm for approximating the geometric median can also be viewed as a special case of IRLS, in which the objective function is the sum of distances of the estimator from the samples.

### Euclidean space

**EuclideanspaceEuclidean 3-space**

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.

### Estimator

**estimatorsestimateestimates**

The geometric median is an important estimator of location in statistics, where it is also known as the L 1 estimator. It is also a standard problem in facility location, where it models the problem of locating a facility to minimize the cost of transportation.

### Location parameter

**locationlocation modelshift parameter**

The geometric median is an important estimator of location in statistics, where it is also known as the L 1 estimator. It is also a standard problem in facility location, where it models the problem of locating a facility to minimize the cost of transportation.

### Facility location

**facilitylocate facilitiesUncapacitated Facility Location**

The geometric median is an important estimator of location in statistics, where it is also known as the L 1 estimator. It is also a standard problem in facility location, where it models the problem of locating a facility to minimize the cost of transportation.

### Steiner tree problem

**Steiner pointminimum Steiner treeSteiner tree**

The special case of the problem for three points in the plane (that is, m = 3 and n = 2 in the definition below) 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 and solved by Evangelista Torricelli.

### Pierre de Fermat

**FermatFermat, Pierre dede Fermat, Pierre**

The special case of the problem for three points in the plane (that is, m = 3 and n = 2 in the definition below) 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 and solved by Evangelista Torricelli.

### Evangelista Torricelli

**TorricelliTorricelli, EvangelistaTorricellian**

The special case of the problem for three points in the plane (that is, m = 3 and n = 2 in the definition below) 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 and solved by Evangelista Torricelli.

### Arg max

**arg minargmaxargument of the maximum**

Here, arg min means the value of the argument y which minimizes the sum.

### Euclidean distance

**EuclideandistanceEuclidean metric**

In this case, it is the point y from where the sum of all Euclidean distances to the x_i's is minimum.

### Univariate

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.

### Line (geometry)

**linelinesstraight line**

The geometric median is unique whenever the points are not collinear.

### Equivariant map

**equivariantintertwining operatorintertwiner**

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.

### Similarity (geometry)

**similarsimilaritysimilar triangles**

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.

### Translation (geometry)

**translationtranslationstranslational**

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.

### Rotation (mathematics)

**rotationrotationsrotate**

### Cartesian coordinate system

**Cartesian coordinatesCartesianaxes**

### Robust statistics

**robustbreakdown pointrobustness**

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.