  Multi-domain translation seeks to learn a probabilistic coupling between
marginal distributions that reflects the correspondence between different
domains. We assume that data from different domains are generated from a shared
latent representation based on a structural equation model. Under this
assumption, we show that the problem of computing a probabilistic coupling
between marginals is equivalent to learning multiple uncoupled autoencoders
that embed to a given shared latent distribution. In addition, we propose a new
framework and algorithm for multi-domain translation based on learning the
shared latent distribution and training autoencoders under distributional
constraints. A key practical advantage of our framework is that new
autoencoders (i.e., new domains) can be added sequentially to the model without
retraining on the other domains, which we demonstrate experimentally on image
as well as genomics datasets.
