  Based on the manifold hypothesis, real-world data often lie on a
low-dimensional manifold, while normalizing flows as a likelihood-based
generative model are incapable of finding this manifold due to their structural
constraints. So, one interesting question arises: $\textit{"Can we find
sub-manifold(s) of data in normalizing flows and estimate the density of the
data on the sub-manifold(s)?"}$. In this paper, we introduce two approaches,
namely per-pixel penalized log-likelihood and hierarchical training, to answer
the mentioned question. We propose a single-step method for joint manifold
learning and density estimation by disentangling the transformed space obtained
by normalizing flows to manifold and off-manifold parts. This is done by a
per-pixel penalized likelihood function for learning a sub-manifold of the
data. Normalizing flows assume the transformed data is Gaussianizationed, but
this imposed assumption is not necessarily true, especially in high dimensions.
To tackle this problem, a hierarchical training approach is employed to improve
the density estimation on the sub-manifold. The results validate the
superiority of the proposed methods in simultaneous manifold learning and
density estimation using normalizing flows in terms of generated image quality
and likelihood.
