I am working with a specific kind of random projection: I have some data that belongs to $\mathbb{R}^{i,j,k}$, and I am using a random projection to only embed the third axis to $p<k$, so that my data now belongs to $\mathbb{R}^{i,j,p}$ after projection. I would like to know how the Johnson-Lindenstrauss lemma would apply in this case as the whole space is not randomly projected but only a subspace of it... Would the distortion $\epsilon$ be the same as if I projected the whole space? Or more? Or less?
Any help is welcome at this point, I've been searching for days ! Thanks a lot and sorry if my notations are'nt strictly rigorous...
