  Real-world geometry and 3D vision tasks are replete with challenging
symmetries that defy tractable analytical expression. In this paper, we
introduce Neural Isometries, an autoencoder framework which learns to map the
observation space to a general-purpose latent space wherein encodings are
related by isometries whenever their corresponding observations are
geometrically related in world space. Specifically, we regularize the latent
space such that maps between encodings preserve a learned inner product and
commute with a learned functional operator, in the same manner as rigid-body
transformations commute with the Laplacian. This approach forms an effective
backbone for self-supervised representation learning, and we demonstrate that a
simple off-the-shelf equivariant network operating in the pre-trained latent
space can achieve results on par with meticulously-engineered, handcrafted
networks designed to handle complex, nonlinear symmetries. Furthermore,
isometric maps capture information about the respective transformations in
world space, and we show that this allows us to regress camera poses directly
from the coefficients of the maps between encodings of adjacent views of a
scene.
