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Context: We therefore seek an algorithm that can learn to translate between domains without paired input-output examples (Figure 2, right). We assume there is some underlying relationship between the domains – for example, that they are two different renderings of the same underlying scene – and seek to learn that relationship. Although we lack supervision in the form of paired examples, we can exploit supervision at the level of sets: we are given one set of images in domain X and a different set in domain Y. We may train a mapping G:X\rightarrow Y such that the output \hat{y}=G(x), x\in X, is indistinguishable from images y\in Y by an adversary trained to classify \hat{y} apart from y. In theory, this objective can induce an output distribution over \hat{y} that matches the empirical distribution p_{data}(y) (in general, this requires G to be stochastic) [16]. The optimal G thereby translates the domain X to a domain \hat{Y} distributed identically to Y. However, such a translation does not guarantee that an individual input x and output y are paired up in a meaningful way – there are infinitely many mappings G that will induce the same distribution over \hat{y}. Moreover, in practice, we have found it difficult to optimize the adversarial objective in isolation: standard procedures often lead to the well-known problem of mode collapse, where all input images map to the same output image and the optimization fails to make progress [15].

Question: What if there is no underlying relationship?
[The translation between domains, with no underlying relationship, will have input x and output y paired up in an unmeaningful way ]