  Representing a manifold of very high-dimensional data with generative models
has been shown to be computationally efficient in practice. However, this
requires that the data manifold admits a global parameterization. In order to
represent manifolds of arbitrary topology, we propose to learn a mixture model
of variational autoencoders. Here, every encoder-decoder pair represents one
chart of a manifold. We propose a loss function for maximum likelihood
estimation of the model weights and choose an architecture that provides us the
analytical expression of the charts and of their inverses. Once the manifold is
learned, we use it for solving inverse problems by minimizing a data fidelity
term restricted to the learned manifold. To solve the arising minimization
problem we propose a Riemannian gradient descent algorithm on the learned
manifold. We demonstrate the performance of our method for low-dimensional toy
examples as well as for deblurring and electrical impedance tomography on
certain image manifolds.
