[dchftcs]

If you preserve the l2 distance you preserve the inner product, that's somewhat tautological in an L2 space. Just that the degree you can preserve inner products can be misleading, main problem is that orthogonal vectors may only become near-orthogonal which is sometimes a big deal, though perfect correlations are preserved because the JL transform is linear. Both can be seen looking at: https://en.wikipedia.org/wiki/Polarization_identity

[srean]

> If you preserve the l2 distance you preserve the inner product

That's trivially untrue. You can move the origin around and that doesn't change the el_2 metric but will change the inner product.

This would not happen for random rotations of course because they do not change the origin. However random Euclidean motions can change the origin.

[dchftcs]

Right, indeed you need to first preserve the origin, but also that is trivially true for a linear map like JL.

[srean]

As far as I can recall JL holds for affine transformations too, in any case it's an existence result. Have to double check on the affine bit.

The popular proof does uses random linear transforms and they indeed will not change the origin, but that's just one class of transforms with the JL property.