Atlas Generative Models and Geodesic Interpolation

Jakob Stolberg-Larsen, Stefan Sommer

Generative neural networks have a well recognized ability to estimate underlying manifold structure of high dimensional data. However, if a simply connected latent space is used, it is not possible to faithfully represent a manifold with non-trivial homotopy type. In this work we define the general class of Atlas Generative Models (AGMs), models with hybrid discrete-continuous latent space that estimate an atlas on the underlying data manifold together with a partition of unity on the data space. We identify existing examples of models from various popular generative paradigms that fit into this class. Due to the atlas interpretation, ideas from non-linear latent space analysis and statistics, e.g. geodesic interpolation, which has previously only been investigated for models with simply connected latent spaces, may be extended to the entire class of AGMs in a natural way. We exemplify this by generalizing an algorithm for graph based geodesic interpolation to the setting of AGMs, and verify its performance experimentally.

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