Scaling to vector unit length

When scaling to vector unit length, we scale individual samples or observations so that the transformed vector has a length of 1, or in other words, a norm of 1. Note that this scaling technique scales each individual observation and not each individual variable. To be clear, in all the scaling methods that we discussed so far in the chapter, the algorithms learned some parameters from each variable and then used those parameters to shift or rescale the distribution of the variables. On the contrary, when we scale to the unit length, we seek to normalize each observation individually, contemplating their values across all features.

Scaling to the unit norm is achieved by dividing each observation vector by either the Manhattan distance (l1 norm) or the Euclidean distance (l2 norm) of the vector. The Manhattan distance is given by the sum of the absolute components of the vector:

The Euclidean distance is given by the square...