Manifold Learning in Quotient Spaces

When learning 3D shapes we are usually interested in their intrinsic geometry rather than in their orientation. To deal with the orientation variations the usual trick consists in augmenting the data to exhibit all possible variability, and thus let the model learn both the geometry as well as the rotations. In this paper we introduce a new autoencoder model for encoding and synthesis of 3D shapes. To get rid of undesirable input variability our model learns a manifold in a quotient space of the input space. Typically, we propose to quotient the space of 3D models by the action of rotations. Thus, our quotient autoencoder allows to directly learn in the space of interest, ignoring side information. This is reflected in better performances on reconstruction and interpolation tasks, as our experiments show that our model outperforms a vanilla autoencoder on the well-known Shapenet dataset. Moreover, our model learns a rotation-invariant representation, leading to interesting results in shapes co-alignment. Finally, we extend our quotient autoencoder to quotient by non-rigid transformations.