Cosine similarity

In data analysis, cosine similarity is a measure of similarity between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided by the product of their lengths. It follows that the cosine similarity does not depend on the magnitudes of the vectors, but only on their angle. The cosine similarity always belongs to the interval ${\displaystyle [-1,1].}$ For example, two proportional vectors have a cosine similarity of 1, two orthogonal vectors have a similarity of 0, and two opposite vectors have a similarity of -1. In some contexts, the component values of the vectors cannot be negative, in which case the cosine similarity is bounded in ${\displaystyle [0,1]}$.

For example, in information retrieval and text mining, each word is assigned a different coordinate and a document is represented by the vector of the numbers of occurrences of each word in the document. Cosine similarity then gives a useful measure of how similar two documents are likely to be, in terms of their subject matter, and independently of the length of the documents.^[1]

The technique is also used to measure cohesion within clusters in the field of data mining.^[2]

One advantage of cosine similarity is its low complexity, especially for sparse vectors: only the non-zero coordinates need to be considered.

Other names for cosine similarity include Orchini similarity and Tucker coefficient of congruence; the Otsuka–Ochiai similarity (see below) is cosine similarity applied to binary data.