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Measuring the similarity between data points often requires domain knowledge,which can in parts be compensated by relying on unsupervised methods such aslatent-variable models, where similarity/distance is estimated in a morecompact latent space. Prevalent is the use of the Euclidean metric, which hasthe drawback of ignoring information about similarity of data stored in thedecoder, as captured by the framework of Riemannian geometry. We propose anextension to the framework of variational auto-encoders allows learning flatlatent manifolds, where the Euclidean metric is a proxy for the similaritybetween data points. This is achieved by defining the latent space as aRiemannian manifold and by regularising the metric tensor to be a scaledidentity matrix. Additionally, we replace the compact prior typically used invariational auto-encoders with a recently presented, more expressivehierarchical one---and formulate the learning problem as a constrainedoptimisation problem. We evaluate our method on a range of data-sets, includinga video-tracking benchmark, where the performance of our unsupervised approachnears that of state-of-the-art supervised approaches, while retaining thecomputational efficiency of straight-line-based approaches.

Learning Flat Latent Manifolds with VAEs