Cosine Similarity

Overview Cosine Similarity is the most common metric used to measure how similar two documents (vectors) are, irrespective of their size. It measures the cosine of the angle between two vectors projected in a multi-dimensional space.

Formula $$ ext{similarity} = cos( heta) = frac{A cdot B}{|A| |B|}$$

• Range: -1 to 1.
• 1: Vectors point in exactly same direction (Identical meaning).
• 0: Vectors are orthogonal (90 degrees, Unrelated).
• -1: Vectors are opposite (180 degrees, Opposite meaning).

Why not Euclidean Distance? Euclidean distance measures the magnitude.

• Document A: "I like app."
• Document B: "I like app. I like app. I like app."
• Euclidean: Far apart (B is much longer).
• Cosine: Identical (Same angle/topic).

For text search, we usually care about the topic (angle), not the frequency (length), making Cosine Similarity superior.

Optimization If vectors are normalized (length = 1), then $|A| = 1$ and $|B| = 1$. The formula simplifies to just the Dot Product ($A cdot B$), which is extremely fast to compute on hardware.