  Despite the success of Generative Adversarial Networks (GANs), their training
suffers from several well-known problems, including mode collapse and
difficulties learning a disconnected set of manifolds. In this paper, we break
down the challenging task of learning complex high dimensional distributions,
supporting diverse data samples, to simpler sub-tasks. Our solution relies on
designing a partitioner that breaks the space into smaller regions, each having
a simpler distribution, and training a different generator for each partition.
This is done in an unsupervised manner without requiring any labels.
  We formulate two desired criteria for the space partitioner that aid the
training of our mixture of generators: 1) to produce connected partitions and
2) provide a proxy of distance between partitions and data samples, along with
a direction for reducing that distance. These criteria are developed to avoid
producing samples from places with non-existent data density, and also
facilitate training by providing additional direction to the generators. We
develop theoretical constraints for a space partitioner to satisfy the above
criteria. Guided by our theoretical analysis, we design an effective neural
architecture for the space partitioner that empirically assures these
conditions. Experimental results on various standard benchmarks show that the
proposed unsupervised model outperforms several recent methods.
