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Autoencoders, which consist of an encoder and a decoder, are widely used inmachine learning for dimension reduction of high-dimensional data. The encoderembeds the input data manifold into a lower-dimensional latent space, while thedecoder represents the inverse map, providing a parametrization of the datamanifold by the manifold in latent space. A good regularity and structure ofthe embedded manifold may substantially simplify further data processing taskssuch as cluster analysis or data interpolation. We propose and analyze a novelregularization for learning the encoder component of an autoencoder: a lossfunctional that prefers isometric, extrinsically flat embeddings and allows totrain the encoder on its own. To perform the training it is assumed that forpairs of nearby points on the input manifold their local Riemannian distanceand their local Riemannian average can be evaluated. The loss functional iscomputed via Monte Carlo integration with different sampling strategies forpairs of points on the input manifold. Our main theorem identifies a geometricloss functional of the embedding map as the $\Gamma$-limit of thesampling-dependent loss functionals. Numerical tests, using image data thatencodes different explicitly given data manifolds, show that smooth manifoldembeddings into latent space are obtained. Due to the promotion of extrinsicflatness, these embeddings are regular enough such that interpolation betweennot too distant points on the manifold is well approximated by linearinterpolation in latent space as one possible postprocessing.

Convergent autoencoder approximation of low bending and low distortion manifold embeddings