Equation 1 · Part 6 · Embeddings and the Geometry of Similarity
Denominator: lVert u rVert lVert v rVert
What this part means
The complete quantity below the fraction bar; it must be nonzero for this division.
Its job in the formula
lVert u rVert lVert v rVert occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Full expression→Denominator: lVert u rVert lVert v rVert→Article meaning
The passage around this formula
The conventional similarity metric is the cosine of the angle between two vectors: . Cosine became conventional for reasons that are mostly good. It is scale-invariant, which matters when vector norms correlate with nuisance properties like token frequency or document length. It reduces to an inner product on normalised vectors, which is cheap and which most approximate indexes support natively. And it is what several influential embedding models were explicitly trained to make meaningful: Sentence-BERT fine-tuned siamese networks precisely so that sentence embeddings could be compared with cosine similarity, cutting a pairwise-comparison workload from roughly 65 hours to…
Learn the underlying idea
A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.
Open the illustrated fractions: division written vertically guide →
Sources cited in the surrounding passage
These citations provide research context; check each source for the exact claim it supports.