Defining an action of SO(d)-rotations on images generated by projections of d-dimensional objects: Applications to pose inference with Geometric VAEs

TL;DR

This paper investigates the application of $SO(d)$-rotations in geometric VAEs for pose inference, revealing that defining a group action on image data requires specific geometric constraints on the underlying volume.

Contribution

It demonstrates that the assumption of data lying on a Lie group manifold often fails without geometric constraints, impacting pose inference in geometric VAEs.

Findings

Group action on image data generally fails without geometric constraints

Proper pose inference depends on specific geometric constraints

Experiments confirm the importance of these constraints for accurate inference

Abstract

Recent advances in variational autoencoders (VAEs) have enabled learning latent manifolds as compact Lie groups, such as $SO(d)$. Since this approach assumes that data lies on a subspace that is homeomorphic to the Lie group itself, we here investigate how this assumption holds in the context of images that are generated by projecting a $d$-dimensional volume with unknown pose in $SO(d)$. Upon examining different theoretical candidates for the group and image space, we show that the attempt to define a group action on the data space generally fails, as it requires more specific geometric constraints on the volume. Using geometric VAEs, our experiments confirm that this constraint is key to proper pose inference, and we discuss the potential of these results for applications and future work.