  Transformation-based methods have been an attractive approach in
non-parametric inference for problems such as unconditional and conditional
density estimation due to their unique hierarchical structure that models the
data as flexible transformation of a set of common latent variables. More
recently, transformation-based models have been used in variational inference
(VI) to construct flexible implicit families of variational distributions.
However, their use in both non-parametric inference and variational inference
lacks theoretical justification. We provide theoretical justification for the
use of non-linear latent variable models (NL-LVMs) in non-parametric inference
by showing that the support of the transformation induced prior in the space of
densities is sufficiently large in the $L_1$ sense. We also show that, when a
Gaussian process (GP) prior is placed on the transformation function, the
posterior concentrates at the optimal rate up to a logarithmic factor. Adopting
the flexibility demonstrated in the non-parametric setting, we use the NL-LVM
to construct an implicit family of variational distributions, deemed GP-IVI. We
delineate sufficient conditions under which GP-IVI achieves optimal risk bounds
and approximates the true posterior in the sense of the Kullback-Leibler
divergence. To the best of our knowledge, this is the first work on providing
theoretical guarantees for implicit variational inference.
