Cosine - Embeddings

Cosine similarity adds vector lengths. Squared norms are exact, while the square-root lengths are named instead of decimalized, so cosine exposes the boundary.

highlighted = computed this step

Cosine is the named boundary

Cosine divides a dot product by the two vector lengths. The squared norms are exact dot products of a vector with itself, but each length introduces a square root. That square root is the named boundary of this book.

\cos(u,v)=\frac{u\cdot v}{\|u\|\|v\|},\quad \|u\|=\sqrt{u\cdot u}

Exact squared norms

The squared norms are cat dot cat = 10, kitten dot kitten = 5, and car dot car = 10. Therefore cosine(cat, kitten) is named as 7 over sqrt(10) times sqrt(5). The numerator is exact; the length magnitudes stay symbolic.

\frac{7}{\sqrt{10}\sqrt{5}}

Summary

The dot products stay exact integers. The square-root length is the named non-exact step here, so the diagram renders 7/(sqrt(10)*sqrt(5)) as a symbol and never as a decimal. Exact dot, named magnitude: that is the register split.

\text{named }\sqrt{\ } \text{ boundary; no decimal cosine}