What is the \(\textsf{L}_1\) Metric?

It is a case of the general Minkowski metric:

\[ d_k(\mathbf{x},\mathbf{y}) = \left( \sum_i^m| x_i - y_i |^k \right)^\frac{1}{k} \] For \(k=1\) - sometimes referred to as the city block metric or the Manhatten metric, particularly if \(\mathbf{x}\) and \(\mathbf{y}\) are 2D vectors (that is, \(m=2\)), since it approximates road distances when the roads form a regular mesh, such as those in Manhatten. Another common case of the metric is where \(k=2\), where the metric is simply Euclidean distance. A notable property of Euclidean distance is that it is rotation invariant - it may easily be verified that that when \(m=2\), if \(\mathbf{R}_\theta\) is a rotation matrix through angle \(\theta\), so we may write

\[ \mathbf{R}_\theta = \left( \begin{array}{rr} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right) \]

then \[ d_2(\mathbf{x},\mathbf{y}) = d_2(\mathbf{R}_\theta \mathbf{x},\mathbf{R}_\theta \mathbf{y} ) \]

However this is not generally the case:

\[ d_k(\mathbf{x},\mathbf{y}) \ne d_k(\mathbf{R}_\theta \mathbf{x},\mathbf{R}_\theta \mathbf{y} ) \textsf{ if } k \ne 2 \]

In particular \(d_1(.,.)\) is not rotation invariant.

Kernels based on distances other than \(d_2(.,.)\)

Suppose we wished to impliment a 2D kernel function (for example to use in kernel density estimation or geographically weighted regression) based on the \(\textsf{L}_1\) metric. A symmetrical 2D kernel generally takes the form

\[ \frac{1}{h^2}K\left(\frac{d(\mathbf{x},\mathbf{y})}{h}\right) \]

Where typically \(K\) takes an argument in the range \([0,\infty)\) and usually

Common examples are

where \(I_{s}\) is 1 if \(s\) is true, and 0 otherwise. Typically, distances are based on \(k=2\) but could be defined otherwise.
Below is an interactive, rotatable and zoomable 3D image of the kernel derived when \(K(d) = \exp(-d)\) and \(d(.,.) = d_1(.,.)\).