Nicely written and very approachable. Might add a paragraph on why to use cosine similarity, as it gives a chance to illustrate how the n-dimensional vector embedding is used.
If all your vectors have the same 2-norm, the cosine similarity of two vectors is a monotonic function of their euclidean distance (i.e. the minimum in one measure will be the minimum in the other).
If all your vectors don't have the same norm, where's the sense in projecting them onto the unit sphere and measuring the angle between them? You're throwing away important information about the embedding.
Could anyone explain?
In semantic search the magnitude of vectors isn’t relevant - if you needed a measure that took magnitude into account than Euclidean would make sense. E.g. image embeddings based on pixel intensity.
There is other aspect: if all vectors are normalized then cosine distance is just dot product. It is faster to calculate than euclidean distance.
Additionally, the scale of q (the query) is constant across any given search, so might as well normalize q; the results won't change.
Then the only question is whether the scale of v matters... It, again, doesn't tell us anything about the relationship between q and v. So, might as well normalize... And then we are just doing cosine similarity.
Think of partitioning the space by taking random hyperplanes. This can be computed by computing the dot product of the normal to the plane with any given point to see which side of the plane the point is on.
For two points that are very close to each other in Euclidean distance, the chance that a random hyperplane gets right in the middle of them is very low. This is, of course, the angle between as measured from the origin, but that angle gets very small the closer the two points are to each other, in Euclidean norm, and the further they are from the origin. By bisecting the space randomly and seeing which side points are on, we get a decent hashing function, binning similar points together.
Now, a point could be very close the origin and have a small angle to another point very far from the origin but, in some sense, there needs to be a conspiracy for this to happen. If you assume some kind of random distribution of points in D dimensional space, drawn in some sort of uniform like distribution in the D dimensional hyper cube or hyper sphere, the chances that a point will be very near the origin go down, as does the chance it'll lie in a thin cone within some other point.
I believe the basic assumption here is that independent features will be randomly distributed where as correlated features will be clumped together (in Euclidean distance). The randomness pushes unrelated points away from the origin and gives decreasing probability that the angle between them is low.
If points aren't random enough, maybe many of them are very close to the origin while others aren't, or maybe there's some other dependence that causes problems, then this could foil the hyperplane trick.
I believe I heard this basic argument from a talk on locality-sensitive hashing [0]. I think I have the basic intuition, but maybe that talk will be more enlightening.