Distance Metric Space

A Distance Metric Space is a 2-tuple [[math]D,d[/math]] consisting of a vector space [math]D[/math] and a corresponding distance function [math]d[/math] defined over every vector pair.

• Context:
  • It can be a Normed Vector Space.
    • an L1 Norm Metric Space (with an L1 Norm Distance Function).
    • an L2 Norm Metric Space (with an L2 Norm Distance Function).
    • a p Norm Metric Space.
  • It can range from being a Bounded Metric Space (if there is a maximum distance between two points in the space) to being an Unbounded Metric Space.
  • It can range from being a Binary Vector Space, to being an Integer Vector Space, to being a Continuous Vector Space.
  • It can be associated with a Cluster.
  • It can be an Input to a Similarity Search Task, Metric Space Optimization Task.
  • …
• Example(s):
  • an Euclidean Space (with Euclidean Distance Function).
  • a Real Number Space (with Distance Function d(x,y) = |y-x|
  • a Graph Space, (with Graph Distance Function of shortest path between two Graph Nodes.
  • a String Space.
• See: Affine Space, Integer Vector Space, Dimension, Set Measure Space, Topology Space.

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References

2012

• (Wikipedia, 2012) ⇒ http://en.wikipedia.org/wiki/Metric_space
  • QUOTE: In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

    The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of "metric" is a generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them.

    Non-intuitive metric spaces occur in elliptic geometry and hyperbolic geometry. For example, the hyperboloid model of hyperbolic geometry is used in special relativity for a metric space of velocities.

    A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.

• http://en.wikipedia.org/wiki/Metric_space#Definition
  • A metric space is an ordered pair [math](M,d)[/math] where [math]M[/math] is a set and [math]d[/math] is a metric on [math]M[/math], i.e., a function :[math]d \colon M \times M \rightarrow \mathbb{R}[/math] such that for any [math]x, y, z \in M[/math], the following holds:
    1. [math]d(x,y) \ge 0[/math] (non-negative),
    2. [math]d(x,y) = 0\,[/math] iff [math]x = y\,[/math] (identity of indiscernibles),
    3. [math]d(x,y) = d(y,x)\,[/math] (symmetry) and
    4. [math]d(x,z) \le d(x,y) + d(y,z)[/math] (triangle inequality) .
  • The first condition follows from the other three, since: : [math]2d(x,y) = d(x,y) + d(y,x) \ge d(x,x) = 0.[/math] The function [math]d[/math] is also called distance function or simply distance. Often, [math]d[/math] is omitted and one just writes [math]M[/math] for a metric space if it is clear from the context what metric is used.

2009

• (Wordnet, 2009) ⇒ http://wordnet.princeton.edu/perl/webwn
  • a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the ...

• http://en.wiktionary.org/wiki/metric_space
  • Any space whose elements are points, and between any two of which a non-negative real number can be defined as the distance between the points …

2003

• (van Wijk, 2003) ⇒ Jarke J. van Wijk. (2003). “Image based Flow Visualization for Curved Surfaces.” In: Visualization, (VIS 2003).
  • QUOTE: … This requires that the distortion from parametric space to geometric space is taken into account to achieve a … Now that we can produce texture aligned with vector fields on curved surfaces, we consider various... A flow field is defined by the superposition of a linear flow field and a...

1955

• (Kelley, 1955) ⇒ John L Kelley. (1955). “General Topology. D. van Nostrand Company.

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