  Generative adversarial networks (GANs), a class of distribution-learning
methods based on a two-player game between a generator and a discriminator, can
generally be formulated as a minmax problem based on the variational
representation of a divergence between the unknown and the generated
distributions. We introduce structure-preserving GANs as a data-efficient
framework for learning distributions with additional structure such as group
symmetry, by developing new variational representations for divergences. Our
theory shows that we can reduce the discriminator space to its projection on
the invariant discriminator space, using the conditional expectation with
respect to the sigma-algebra associated to the underlying structure. In
addition, we prove that the discriminator space reduction must be accompanied
by a careful design of structured generators, as flawed designs may easily lead
to a catastrophic "mode collapse" of the learned distribution. We contextualize
our framework by building symmetry-preserving GANs for distributions with
intrinsic group symmetry, and demonstrate that both players, namely the
equivariant generator and invariant discriminator, play important but distinct
roles in the learning process. Empirical experiments and ablation studies
across a broad range of data sets, including real-world medical imaging,
validate our theory, and show our proposed methods achieve significantly
improved sample fidelity and diversity -- almost an order of magnitude measured
in Fr\'echet Inception Distance -- especially in the small data regime.
