Some embedding models are explicitly trained on cosine similarity. Otherwise, if you have a 512D vector, discarding magnitude is like discarding just a single dimension (i.e. you get 511 independent dimensions).
This is not quite right; you are actually losing information about each of the dimensions and your mental model of reducing the dimensionality by one is misleading.
Consider [1,0] and [x,x]
Normalised we get [1,0] and [sqrt(.5),sqrt(.5)] — clearly something has changed because the first vector is now larger in dimension zero than the second, despite starting off as an arbitrary value, x, which could have been smaller than 1. As such we have lost information about x’s magnitude which we cannot recover from just the normalized vector.
Well, depends. For some models (especially two tower style models that use a dot product), you're definitely right and it makes a huge difference. In my very limited experience with LLM embeddings, it doesn't seem to make a difference.
Magnitude is not a dimension, it’s information about each value that is lost when you normalize it. To prove this normalize any vector and then try to de-normalize it again.
Magnitude is a dimension. Any 2-dimensional vector can be explicitly transformed into the polar (r, theta) coordinate system where one of the dimensions is magnitude. Any 3-dimensional vector can be transformed into the spherical (r, theta, phi) coordinate where one of the dimensions is magnitude. This is high school mathematics. (Okay I concede that maybe the spherical coordinate system isn't exactly high school material, then just think about longitude, latitude, and distance from the center.)
There's something wrong with the picture here but I can't put my finger on it because my mathematical background here is too old. The space of k dimension vectors all normalized isn't a vector space itself. It's well-behaved in many ways but you lose the 0 vector (may not be relevant). Addition isn't defined anymore, and if you try to keep it inside by normalization post addition, distribution becomes weird. I have no idea what this transformation means for word2vec and friends.
But the intuitive notion is that if you take all 3D and flatten it / expand it to be just the surface of the 3D sphere, then paste yourself onto it Flatland style, it's not the same as if you were to Flatland yourself into the 2D plane. The obvious thing is that triangles won't sum to 180, but also parallel lines will intersect, and all sorts of differing strange things will happen.
I mean, it might still work in practice, but it's obviously different from some method of dimensionality reduction because you're changing the curvature of the space.
The space of all normalized k-dimensional vector is just a unit k-sphere. You can deal with it directly, or you can use the standard inverse stereographic projection to map every point (except for one) onto a plane.
> triangles won't sum to 180
Exactly. Spherical triangles have the sum of their interior angles exceed 180 degrees.
> parallel lines will intersect
Yes because parallel "lines" are really great circles on the sphere.
So is it actually the case that normalizing down and then mapping to the k-1 plane yields a useful (for this purpose) k-1 space? Something feels wrong about the whole thing but I must just have broken intuition.
