  Autoencoders, which consist of an encoder and a decoder, are widely used in
machine learning for dimension reduction of high-dimensional data. The encoder
embeds the input data manifold into a lower-dimensional latent space, while the
decoder represents the inverse map, providing a parametrization of the data
manifold by the manifold in latent space. A good regularity and structure of
the embedded manifold may substantially simplify further data processing tasks
such as cluster analysis or data interpolation. We propose and analyze a novel
regularization for learning the encoder component of an autoencoder: a loss
functional that prefers isometric, extrinsically flat embeddings and allows to
train the encoder on its own. To perform the training it is assumed that for
pairs of nearby points on the input manifold their local Riemannian distance
and their local Riemannian average can be evaluated. The loss functional is
computed via Monte Carlo integration with different sampling strategies for
pairs of points on the input manifold. Our main theorem identifies a geometric
loss functional of the embedding map as the $\Gamma$-limit of the
sampling-dependent loss functionals. Numerical tests, using image data that
encodes different explicitly given data manifolds, show that smooth manifold
embeddings into latent space are obtained. Due to the promotion of extrinsic
flatness, these embeddings are regular enough such that interpolation between
not too distant points on the manifold is well approximated by linear
interpolation in latent space as one possible postprocessing.
