ℓ1 Norm Distance Function

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An ℓ1 norm distance function is a Minkowski distance function with [math]\displaystyle{ d=1 }[/math] (that represents the shortest distance in unit steps along each axis between two points).



References

2015

2011


  • http://en.wikipedia.org/wiki/L1_norm#Formal_description
    • QUOTE: The taxicab distance, [math]\displaystyle{ d_1 }[/math], between two vectors [math]\displaystyle{ \mathbf{p}, \mathbf{q} }[/math] in an n-dimensional real vector space with fixed Cartesian coordinate system, is the sum of the lengths of the projections of the line segment between the points onto the coordinate axes. More formally, [math]\displaystyle{ d_1(\mathbf{p}, \mathbf{q}) = \|\mathbf{p} - \mathbf{q}\|_1 = \sum_{i=1}^n |p_i-q_i|, }[/math] where [math]\displaystyle{ \mathbf{p}=(p_1,p_2,\dots,p_n)\text{ and }\mathbf{q}=(q_1,q_2,\dots,q_n)\, }[/math] are vectors. For example, in the plane, the taxicab distance between [math]\displaystyle{ (p_1,p_2) }[/math] and [math]\displaystyle{ (q_1,q_2) }[/math] is [math]\displaystyle{ | p_1 - q_1 | + | p_2 - q_2 |. }[/math]

      Taxicab distance depends on the rotation of the coordinate system, but does not depend on its reflection about a coordinate axis or its translation. Taxicab geometry satisfies all of Hilbert's axioms (a formalization of Euclidean geometry) except for the side-angle-side axiom, as one can generate two triangles each with two sides and the angle between them the same, and have them not be congruent.


2010

2009

  • (Weisstein, 2009-11-02) ⇒ Eric W. Weisstein. (2009). “L1-Norm." From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/L1-Norm.html
    • A vector norm defined for a vector [math]\displaystyle{ \mathbf{x}=[x_1, x_2, ..., x_n] }[/math], with complex entries by [math]\displaystyle{ |x|_1=\sum_{r=1}^n|x_r| }[/math]. The [math]\displaystyle{ L^1 }[/math]-norm [math]\displaystyle{ |x|_1 }[/math] of a vector [math]\displaystyle{ x }[/math] is ...

2008

1990

  • (Horn & Johnson, 1990) ⇒ R. A. Horn, and C. R. Johnson. (1990). “Norms for Vectors and Matrices." Ch. 5 in Matrix Analysis. Cambridge University Press.