Geometry-Aware Autoencoders for Metric Learning and Generative Modeling on Data Manifolds

Xingzhi Sun, Danqi Liao, Kincaid MacDonald, Yanlei Zhang, Guillaume Huguet, Guy Wolf, Ian Adelstein, Tim G. J. Rudner, Smita Krishnaswamy

Track: Extended abstract

Keywords: Riemannian Generative Modeling, Manifold Learning, Single-cell Dynamics

Abstract: Non-linear dimensionality reduction methods have proven successful at learning low-dimensional representations of high-dimensional point clouds on or near data manifolds. However, existing methods are not easily extensible— for large datasets, it is prohibitively expensive to add new points to these embeddings. As a result, it is very difficult to use existing embeddings generatively, to sample new points on and along these manifolds. In this paper, we propose GAGA (geometry-aware generative autoencoders) a framework which merges the power of generative deep learning with non-linear manifold learning by: 1) learning generalizable geometry-aware neural network embeddings based on non-linear dimensionality reduction methods like PHATE and diffusion maps, 2) deriving a non-euclidean pullback metric on the data space to generate points faithfully along data manifold geodesics, and 3) learning a flow on the manifold that allows us to transport populations. We provide illustration on easily-interpretable synthetic datasets and showcase results on simulated and real single cell datasets. We show that the geodesic-based generation can be especially important for scientific datasets where the manifold represents a state space and geodesics can represent dynamics of entities over this space.

Submission Number: 72