Hands-On Image Processing with Python

Squared Euclidean distance

To compute the nearest neighbors in our dataset, we need to first be able to compute distances between data points. A natural distance function is Euclidean distance; for two vectors x, y ∈ R^d, their Euclidean distance is defined as follows:

Often, we omit the square root and simply compute the squared Euclidean distance. For the purposes of nearest neighbor computations, the two are equivalent: for three vectors x, y, z ∈ R^d, we have ∥x−y∥  ≤  ∥x−z∥ if and only if ∥x−y∥² ≤ ∥x−z∥². Now we just need to be able to compute the squared Euclidean distance.