https://bib.dbvis.de/uploadedFiles/155.pdf

University of Michigan - Ann Arbor

The power parameter p controls which distance metric is used in determining neighbors of a point. The default is p=2, which specifies Euclidean distance. Setting p=1 specifies Manhattan distance (aka city block distance), which calculates distance along a grid-like path. Manhattan distance is preferred for high-dimensional data.

"p" Parameter for KNN Classifier

https://medium.com/@kunal_gohrani/different-types-of-distance-metrics-used-in-machine-learning-e9928c5e26c7#:~:text=Manhattan%20distance%20is%20usually%20preferred,similarity%20between%20two%20data%20points.

Different Types of Distance Metrics used in Machine Learning

Manhattan distance, aka city block distance, is the number of units one must traverse to get from point X to point Y along a grid-like path. In other words, each step involves a change in value across exactly one dimension. In machine learning, Manhattan distance is preferred (over higher values of p) for high-dimensionality data because of simpler computation and because of something called the "curse of high dimensionality."

More formally, let $x_i$ be the value of point X across the $i$th dimension, and let $y_i$ be the value of point Y across the $i$th dimension (NOT to be confused with the conventional (x,y) notation for a point in 2D). Then, in $n$ dimensions, the Manhattan distance between points X and Y is
$d(X,Y) = \sum_{i=1}^n|x_i - y_i|$.

(Note that this is an instantiation of the Minkowski distance formula with p=1.)

Manhattan Distance (p=1)

On the Surprising Behavior of Distance Metrics
in High Dimensional Space

The curse of high dimensionality is the empirical phenomenon that in high dimensional space, concepts of proximity, distance, and/or nearest neighbor lose qualitative meaning. This problem is more severe for Euclidean distance than for Manhattan distance, hence why Manhattan distance is preferred for high dimensional data.

Curse of High Dimensionality

In order to find the Euclidean distance, one has to take the square root of the sum of the squared difference between two points. The formula for calculating the Euclidean distance is given by:

	 $\sqrt{ \sum_{i=1}^k {(x_i - y_i)}^2 }$


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