Euclidean Distance

What Is The Euclidean Distance?

The Euclidean distance between two points in a Euclidean space is the length of a segment between those two points.

Mathematically:

Given two points p and q in a Euclidean space of dimension n, with coordinates (p₁, p₂, … , p_n) and (q₁, q₂, … , q_n), the Euclidean distance between them is defined as:

As this formula and the figure below shows, the Euclidean distance can be calculated via the Pythagorean theorem.

Figure 1: Pythagorean theorem.

Source: Kmhkmh, CC BY 4.0, via Wikimedia Commons

Euclidean distance has many important properties. Among them are:

Symmetry:

Given two points p and q, then d(p,q) = d(q,p)

Positive:

The value of the Euclidean distance between two points is always a positive number.

Triangle Equality:

Given three points p, q and r, then d(p,q) + d(q,r) ≥ d(p,r)

In machine learning, the Euclidean distance is just one of the four distance measures used between a pair of samples p and q in an n-dimensional feature space. The other three are the Manhattan distance, the Minkowski distance, and the Hamming distance. Of the four distances, the Euclidean gives the shortest value between any two points.

Why Is The Euclidean Distance Effective?

Many machine learning algorithms use the Euclidean distance to measure the similarity between observations and supervised and unsupervised learning. Commonly known algorithms that use it are the K-nearest neighbors (classification) and K-means (clustering), which estimate distances to find the k-closest points. It is also used in hierarchical clustering and agglomerative clustering to calculate the distances between clusters.

Deep neural networks (DNNs) can calculate Euclidean distances in cases where the direct evaluation process becomes very complicated.