The Riemannian Geometry of Deep Generative Models
This work provides foundational tools for understanding the geometric properties of deep generative models, which could impact researchers in machine learning and AI by offering insights into model behavior and potential improvements.
The paper investigates the Riemannian geometry of manifolds learned by deep generative models, developing algorithms for computing geodesics and parallel translation to analyze distances and generate analogies on these manifolds, with experiments on real image data showing that the manifolds are surprisingly close to zero curvature, implying linear latent paths approximate geodesics.
Deep generative models learn a mapping from a low dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate the Riemannian geometry of these generated manifolds. First, we develop efficient algorithms for computing geodesic curves, which provide an intrinsic notion of distance between points on the manifold. Second, we develop an algorithm for parallel translation of a tangent vector along a path on the manifold. We show how parallel translation can be used to generate analogies, i.e., to transport a change in one data point into a semantically similar change of another data point. Our experiments on real image data show that the manifolds learned by deep generative models, while nonlinear, are surprisingly close to zero curvature. The practical implication is that linear paths in the latent space closely approximate geodesics on the generated manifold. However, further investigation into this phenomenon is warranted, to identify if there are other architectures or datasets where curvature plays a more prominent role. We believe that exploring the Riemannian geometry of deep generative models, using the tools developed in this paper, will be an important step in understanding the high-dimensional, nonlinear spaces these models learn.