Reproduce “Manifold Learning by Mixture Models of VAEs for Inverse Problems” on OpenResearch

About this paper

Giovanni S. Alberti, Johannes Hertrich, Matteo Santacesaria, Silvia Sciutto

This research from the University of Genoa and UCL proposes a method to learn high-dimensional data manifolds with arbitrary topologies by using a mixture model of Variational Autoencoders, where each VAE acts as a local chart. The work also introduces a Riemannian gradient descent algorithm to solve ill-posed inverse problems directly on these learned manifolds, demonstrating improved performance in image deblurring and electrical impedance tomography over single-chart models.

Key insights

• Learning an 'atlas' via a mixture of VAEs effectively approximates high-dimensional data manifolds with complex, unknown topologies, overcoming limitations of single generative models that assume global parameterization.
• Designing VAE architectures with analytically invertible decoders and encoders enables explicit chart formulations and their inverses, crucial for both training stability and subsequent optimization on the learned manifold.
• Applying Riemannian gradient descent directly on the learned manifold, using chart-specific retractions and weighted averaging across charts, provides a robust method for solving inverse problems without requiring a global parameterization of the data.

Summary by alphaXiv.