  Deep generative models have demonstrated successful applications in learning
non-linear data distributions through a number of latent variables and these
models use a nonlinear function (generator) to map latent samples into the data
space. On the other hand, the nonlinearity of the generator implies that the
latent space shows an unsatisfactory projection of the data space, which
results in poor representation learning. This weak projection, however, can be
addressed by a Riemannian metric, and we show that geodesics computation and
accurate interpolations between data samples on the Riemannian manifold can
substantially improve the performance of deep generative models. In this paper,
a Variational spatial-Transformer AutoEncoder (VTAE) is proposed to minimize
geodesics on a Riemannian manifold and improve representation learning. In
particular, we carefully design the variational autoencoder with an encoded
spatial-Transformer to explicitly expand the latent variable model to data on a
Riemannian manifold, and obtain global context modelling. Moreover, to have
smooth and plausible interpolations while traversing between two different
objects' latent representations, we propose a geodesic interpolation network
different from the existing models that use linear interpolation with inferior
performance. Experiments on benchmarks show that our proposed model can improve
predictive accuracy and versatility over a range of computer vision tasks,
including image interpolations, and reconstructions.
