  In this paper, we propose a method to learn a minimizing geodesic within a
data manifold. Along the learned geodesic, our method can generate high-quality
interpolations between two given data samples. Specifically, we use an
autoencoder network to map data samples into latent space and perform
interpolation via an interpolation network. We add prior geometric information
to regularize our autoencoder for the convexity of representations so that for
any given interpolation approach, the generated interpolations remain within
the distribution of the data manifold. Before the learning of a geodesic, a
proper Riemannianmetric should be defined. Therefore, we induce a Riemannian
metric by the canonical metric in the Euclidean space which the data manifold
is isometrically immersed in. Based on this defined Riemannian metric, we
introduce a constant speed loss and a minimizing geodesic loss to regularize
the interpolation network to generate uniform interpolation along the learned
geodesic on the manifold. We provide a theoretical analysis of our model and
use image translation as an example to demonstrate the effectiveness of our
method.
