No, I don’t think the author is saying one dimensional - the vectors are represented by magnitudes in almost all of the embedding dimensions.
They are still a “direction” in the way that [0.5, 0.5] in x,y space is a 45 degree angle, and in that direction it has a magnitude of around 0.7
So of course you could probably define some other vector space where many of the different labeled vectors are translated to magnitudes in the original embedding space, letting you do things like have a “tone” slider.
I think GP is saying that GGP assumes "tone" is one direction, in the sense there exists a vector V representing "tone direction", and you can scale "tone" independently by multiplying that vector with a scalar - hence, 1 dimension.
I'd say this assumption is both right and wrong. Wrong, because it's unlikely there's a direction in embedding space corresponding to a platonic ideal of "tone". Right, because I suspect that, for sufficiently large embedding space (on the order of what goes into current LLMs), any continuous concept we can articulate will have a corresponding direction in the embedding space, that's roughly as sharp as our ability to precisely define the concept.
They are still a “direction” in the way that [0.5, 0.5] in x,y space is a 45 degree angle, and in that direction it has a magnitude of around 0.7
So of course you could probably define some other vector space where many of the different labeled vectors are translated to magnitudes in the original embedding space, letting you do things like have a “tone” slider.