Data Synthesis

Indeed, $G^*({\bf x},T,g)$ is a generative model. To generate from $G^*({\bf x},T,g)$, we modify the method in Section 2.2.2 slightly. Generation from $G^*({\bf x},T,g)$ is most easily thought of as a two-step process [24]

1. Draw a sample from the feature density $g({\bf z})$. This sample is denoted by ${\bf z}^*$.
2. Determine the manifold ${\cal M}({\bf z}^*)$, defined in eq. (2.4), and draw a sample from the manifold using the uniform distribution. We call this uniform manifold sampling (UMS). Note that one must be careful to insure that, when the manifold is parameterized, for example with angle, that the uniform distribution must be imposed in ${\bf x}$, which does not necessarily mean that the parameters will be uniformly distributed.

The above process seems overly simple, but this is deceptive, as we will later see. Not only can the second step be diffiicult, but the process generates remarkably complex distributions.