Cosine Similarity

08/17/2023

Cosine similarity measures the similarity between two vectors by calculating the cosine of the angle between the two vectors.
Why cosine of the angle between \(x\) and \(y\) gives use the similarity?

Cosine function is \(1\) at the \(\theta = 0\) and \(-1\) at \(\theta = 180\). Therefore, if the two vectors overlap more perfectly, the cosine similarity approaches \(1\).

Calculation of cosine of the angle between \(x\) and \(y\) as: \[ \,\\ \cos(\theta) = \frac{x \cdot y}{\|x\| \, \|y\|} \,\\ \]
The above expression comes from the definition of the dot product:
- The dot product of two vectors \(x\) and \(y\) is defined as: \[ \,\\ x \cdot y = \|x\| \, \|y\| \cos(\theta) \,\\ \]
- Rearranging this equation to solve for \(\cos(\theta)\) gives us: \[ \,\\ \cos(\theta) = \frac{x \cdot y}{\|x\| \, \|y\|} \,\\ \]
- This is equivalent to \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\) in a right-angled triangle, where \(\text{adjacent} = x \cdot y \) and \(\text{hypotenuse} = \|x\| \, \|y\| \)