Re-synthesis of from using UMS (Section 3.3)
is done as follows.
Given a fixed feature value
the manifold is given by
Any that meets the first requirement
can be written
and where is the
that spans the linear subspace orthogonal to the columns of .
Thus, spans the manifold.
To satisfy the second requirement, we need
Thus, we need the vector to have length
Thus, lies on a hyper-sphere. Uniformly sampling
a hyper-sphere is accomplished by drawing
as iid samples of zero-mean Gaussian random
variable, then normalizing to have length equal to the
This works because the multivariate standard Gaussian has a
distribution that projects evenly anywhere on
the standard hyper-sphere.
In summary, the sampling method is:
(1) Draw a sample
, samples of independent
Gaussian samples of mean 0 and variance 1,
then (3) Let
For more information see the function