The Riemannian Geometry of Deep Generative Models

Hang Shao, Abhishek Kumar, P. Thomas Fletcher; Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, 2018, pp. 315-323


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.

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[pdf] [arXiv]
author = {Shao, Hang and Kumar, Abhishek and Thomas Fletcher, P.},
title = {The Riemannian Geometry of Deep Generative Models},
booktitle = {Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops},
month = {June},
year = {2018}