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

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

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.