  Flow-based generative models are composed of invertible transformations
between two random variables of the same dimension. Therefore, flow-based
models cannot be adequately trained if the dimension of the data distribution
does not match that of the underlying target distribution. In this paper, we
propose SoftFlow, a probabilistic framework for training normalizing flows on
manifolds. To sidestep the dimension mismatch problem, SoftFlow estimates a
conditional distribution of the perturbed input data instead of learning the
data distribution directly. We experimentally show that SoftFlow can capture
the innate structure of the manifold data and generate high-quality samples
unlike the conventional flow-based models. Furthermore, we apply the proposed
framework to 3D point clouds to alleviate the difficulty of forming thin
structures for flow-based models. The proposed model for 3D point clouds,
namely SoftPointFlow, can estimate the distribution of various shapes more
accurately and achieves state-of-the-art performance in point cloud generation.
