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

where
and where is the
ortho-normal matrix
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
hyper-sphere radius.
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,
(2) Let
then (3) Let
For more information see the function
`software/module_lin_gauss_synth.m`.

Baggenstoss
2017-05-19