  Noting the importance of the latent variables in inference and learning, we
propose a novel framework for autoencoders based on the homeomorphic
transformation of latent variables, which could reduce the distance between
vectors in the transformed space, while preserving the topological properties
of the original space, and investigate the effect of the latent space
transformation on learning generative models and denoising corrupted data. The
experimental results demonstrate that our generative and denoising models based
on the proposed framework can provide better performance than conventional
variational and denoising autoencoders due to the transformation, where we
evaluate the performance of generative and denoising models in terms of the
Hausdorff distance between the sets of training and processed i.e., either
generated or denoised images, which can objectively measure their differences,
as well as through direct comparison of the visual characteristics of the
processed images.
