Cosine similarity

Cosine similarity measures how closely two nonzero vectors point in the same direction, using their dot product divided by the product of their lengths. It is unaffected by vector magnitude.

Dot product: An operation on two vectors that sums the products of corresponding components. It supplies the numerator that measures directional alignment.
Information retrieval: The process of finding relevant documents or information in a collection. Search systems rank documents by the directional match between query and document vectors.
Pearson correlation coefficient: A statistic measuring linear association between two variables after centering them by their means. Centering distinguishes correlation from cosine similarity on the original vectors.
L2 normalization: Scaling a nonzero vector to have Euclidean length one. It makes dot-product comparisons equivalent to cosine similarity.
Vector norm: A function that assigns a nonnegative length to a vector. The two vector lengths form the denominator and remove magnitude effects.
Vector space model: A representation of documents and queries as weighted term vectors. Cosine similarity compares their term-weight directions despite differing document lengths.
Jaccard index: A similarity measure equal to the intersection size divided by the union size of two sets. It compares set overlap rather than the angle between numeric vectors.
Sparse matrix: A matrix in which most entries are zero. Text vectors are often sparse, allowing efficient cosine calculations that skip zero entries.
Angle between vectors: The geometric angle between two nonzero vectors, defined through their dot product and lengths. Cosine similarity equals the cosine of this angle.
Term frequency–inverse document frequency: A weighting scheme that scores terms by document frequency and rarity across a collection. Its weights commonly form the vectors compared in document retrieval.

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