# Cosine distance¶

The Cosine distance is a measure of distance between two non-zero vectors of an inner product space.

 Operation Computational methods dense dense

## Mathematical formulation¶

### Computing¶

Given a set $$U$$ of $$n$$ feature vectors $$u_1 = (u_{11}, \ldots, u_{1k}), \ldots, u_n = (u_{n1}, \ldots, u_{nk})$$ of dimension $$k$$ and a set $$V$$ of $$m$$ feature vectors $$v_1 = (v_{11}, \ldots, v_{1k}), \ldots, v_m = (v_{m1}, \ldots, v_{mk})$$ of dimension $$k$$, the problem is to compute the Cosine distance $$D_{cos}(u_i, v_j)$$ for any pair of input vectors:

$D_{cos}(u_i, v_j) = 1 - \frac{\sum_{l=1}^{k}u_{il}v_{jl}}{\sqrt{\sum_{l=1}^{k}u_{il}^{2}}\sqrt{\sum_{l=1}^{k}v_{jl}^{2}}},$

where $$\quad 1 \leq i \leq n, \quad 1 \leq j \leq m$$.

### Computation method: dense¶

The method defines Cosine distance metric, which is used in other algorithms for the distance computation. There is no separate computation mode to compute the distance manually.

## Programming Interface¶

Refer to API Reference: Cosine distance.