CycleGAN can map between two different distributions $X$ and $Y$ with cycle consistency. This is done with generator functions $F: X \mapsto Y$ and $G: Y \mapsto X$, such that $||G(F(x)) - x||_1 \approx 0$, where $x \in X$, and $||F(G(y)) - y||_1 \approx 0$, where $y \in Y$.
What if instead, I want to have a function $H: X \mapsto X$, such that:
- $||H(H(x)) - x||_1 \approx 0$
- But... $||H(x) - x|| > k$, where $k$ is some minimum distance.
So the second point is key here. In plain English, I want to map $x_1 \in X$ to $x_2 \in X$ such that $x_1$ and $x_2$ look like different examples of the data distribution. I'm not so sure the formulation I provided matches this plain English description, but the latter is what I'm really asking.
Also, I'm not sure if cycle consistency is totally needed for what I'm asking, but I think it might be. That's because I want $x_2$ to have the same arrangement of high level objects as $x_1$, but with different finer grain features.
Finally, to give you a more visual representation of what I'm looking for: the following image (taken from the CycleGAN paper) shows oranges translated to apples in the same arrangement.
I'd like to take a dataset of maybe 10 different types of spherical fruits and treat it all as my $X$, and get a function $H$ which can take a picture of an arrangement of say apples, and then produce a similar spatial arrangement of another fruit in the dataset.