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The density px is tractable if the density pz is tractable and the determinant
of the Jacobian of g−1 is tractable. In other words, a simple distribution over
z combined with a transformation g that warps space in complicated ways
can yield a complicated distribution over x, and if g is carefully designed, the
density is tractable too. Models with nonlinear g functions date back at least
to Deco and Brauer (1995). The latest member of this family is real NVP
(Dinh et al., 2016). See figure 10 for some visualizations of ImageNet samples
generated by real NVP. The main drawback to nonlinear ICA models is that
they impose restrictions on the choice of the function g. In particular, the
invertibility requirement means that the latent variables z must have the same
dimensionality as x. GANs were designed to impose very few requirements on
g, and, in particular, admit the use of z with larger dimension than x.