  Machine learning often aims to produce latent embeddings of inputs which lie
in a larger, abstract mathematical space. For example, in the field of 3D
modeling, subsets of Euclidean space can be embedded as vectors using implicit
neural representations. Such subsets also have a natural algebraic structure
including operations (e.g., union) and corresponding laws (e.g.,
associativity). How can we learn to "union" two sets using only their latent
embeddings while respecting associativity? We propose a general procedure for
parameterizing latent space operations that are provably consistent with the
laws on the input space. This is achieved by learning a bijection from the
latent space to a carefully designed mirrored algebra which is constructed on
Euclidean space in accordance with desired laws. We evaluate these structural
transport nets for a range of mirrored algebras against baselines that operate
directly on the latent space. Our experiments provide strong evidence that
respecting the underlying algebraic structure of the input space is key for
learning accurate and self-consistent operations.
