Atlas Generative Models and Geodesic Interpolation
This work addresses a limitation in generative modeling for data with complex topologies, offering a method to apply non-linear analysis techniques more broadly, though it appears incremental as it builds on existing paradigms.
The authors tackled the problem of generative models failing to represent manifolds with non-Euclidean topology by introducing Atlas Generative Models (AGMs), a class with hybrid discrete-continuous latent spaces that estimate an atlas on data manifolds, and they extended geodesic interpolation to AGMs, verifying its performance experimentally.
Generative neural networks have a well recognized ability to estimate underlying manifold structure of high dimensional data. However, if a single latent space is used, it is not possible to faithfully represent a manifold with topology different from Euclidean space. 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.