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

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