UMS

Re-synthesis of $ {\bf x}$ from $ {\bf z}$ using UMS (Section 3.3) is done as follows. Given a fixed feature value $ {\bf z}^* = [{\bf z}_A^*, \; t_2^*]$, the manifold is given by $ {\bf x}: \left\{ {\bf A}^\prime {\bf x}= {\bf z}_A^*, \;\; \sum_{i=1}^N x_i^2=t_2^*
\right\}.$ Any $ {\bf x}$ that meets the first requirement $ {\bf A}^\prime {\bf x}= {\bf z}_A^*$ can be written

$\displaystyle {\bf x}= {\bf x}_A + {\bf B} {\bf u},$

where $ {\bf x}_A={\bf A} ({\bf A}^\prime {\bf A})^{-1} {\bf z}_A^*$ and where $ {\bf B}$ is the $ N\times(N-D)$ ortho-normal matrix that spans the linear subspace orthogonal to the columns of $ {\bf A}$. Thus, $ {\bf u}$ spans the manifold. To satisfy the second requirement, we need $ \vert\vert{\bf x}\vert\vert^2 = \vert\vert{\bf x}_A\vert\vert^2 + \vert\vert{\bf u}\vert\vert^2 = t_2^*.$ Thus, we need the vector $ {\bf u}$ to have length $ \vert\vert{\bf u}\vert\vert=\sqrt{t_2^*-\vert\vert{\bf x}_A\vert\vert^2}.$ Thus, $ {\bf u}$ lies on a hyper-sphere. Uniformly sampling a hyper-sphere is accomplished by drawing $ {\bf u}$ as $ n-D$ iid samples of zero-mean Gaussian random variable, then normalizing $ {\bf u}$ 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 $ \tilde{{\bf u}}$, $ N-D$ samples of independent Gaussian samples of mean 0 and variance 1, (2) Let $ {\bf u}= \frac{\tilde{{\bf u}}}{\Vert\tilde{{\bf u}}\Vert}
\; \sqrt{t_2^*-\vert\vert{\bf x}_A\vert\vert^2},$ then (3) Let $ {\bf x}= {\bf x}_A + {\bf B} {\bf u}.$ For more information see the function software/module_lin_gauss_synth.m.

Baggenstoss 2017-05-19