[renewiltord]

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.

[kccqzy]

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.

[renewiltord]

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.

[kccqzy]

I do not understand the purpose that you are referring to in this comment or the earlier comment. But it is useful for some purposes.