We discuss the feasibility of the following learning problem: given unmatched samples from two domains and nothing else, learn a mapping between the two, which preserves semantics. Due to the lack of paired samples and without any definition of the semantic information, the problem might seem ill-posed. Specifically, in typical cases, it seems possible to build infinitely many alternative mappings  from every target mapping. This apparent ambiguity stands in sharp contrast to the recent empirical success in solving this problem.

 We identify the abstract notion of aligning two domains in a semantic way with concrete terms of minimal relative complexity. A theoretical framework for measuring the complexity of compositions of functions is developed in order to show that it is reasonable to expect the minimal complexity mapping to be unique. The measured complexity used is directly related to the depth of the neural networks being learned and a semantically aligned mapping could then be captured simply by learning using architectures that are not much bigger than the minimal architecture.

 Various predictions are made based on the hypothesis that semantic alignment can be captured by the minimal mapping. These are verified extensively. In addition, a new mapping algorithm is proposed and shown to lead to better mapping results.
Our hypothesis is that given two domains, the lowest complexity mapping that has a low discrepancy approximates the target mapping. The paper addresses the problem of learning mappings between different domains without any supervision, stating three conjectures. Demonstrates that in unsupervised learning on unaligned data it is possible to learn the between domains mapping using GAN only without a reconstruction loss.