  We propose and study the problem of distribution-preserving lossy
compression. Motivated by recent advances in extreme image compression which
allow to maintain artifact-free reconstructions even at very low bitrates, we
propose to optimize the rate-distortion tradeoff under the constraint that the
reconstructed samples follow the distribution of the training data. The
resulting compression system recovers both ends of the spectrum: On one hand,
at zero bitrate it learns a generative model of the data, and at high enough
bitrates it achieves perfect reconstruction. Furthermore, for intermediate
bitrates it smoothly interpolates between learning a generative model of the
training data and perfectly reconstructing the training samples. We study
several methods to approximately solve the proposed optimization problem,
including a novel combination of Wasserstein GAN and Wasserstein Autoencoder,
and present an extensive theoretical and empirical characterization of the
proposed compression systems.
