Let $x_i$ be $N$ points distributed in the interval $[0, d]$ calculated using the equation $$x_i = (s + i \cdot d \cdot \Phi) \mod d$$ where $s$ is a constant describing a starting point, $d$ a constant describing the interval length and $\Phi$ is the Golden Ratio with $\Phi = \frac{1}{2}(1+\sqrt{5})$.
Now lets suppose we wouldn't know the value of $\Phi$ but we are given the $x_i$ out of order and with some small perturbations. If there was no modulo operation, the equation would be something nice and linear and linear regression would produce the best approximation for $\Phi$. However I don't know how regression can be applied with the modulo operation being there. As far as I understand it, there needs to be an inverse function, which is a more complicated thing with modular functions.
Background (you can skip this):
The equation above can be used to create something called Golden Sets (see this paper), which have nice properties if you want to simulate particle systems for example. However they are a bit too regular for my use case. I envision a method that creates point sets that look more like blue noise or more specifically a distribution of points after several cycles of a relaxation method was performed on the particles.
I was wondering if the GRS method can be modified by 1.) using a different constant and 2.) by using more than one in some fashion (this obviously needs more thought). This question is the first hurdle to see if something like this is even possible. If I can perform regression on the data resulting from the relaxation method and get a result that is better than the pure GRS method, this would already reduce the computation necessary for initialization.
Update: I have since found a solution that works for this specific case, but I consider the original question unsolved as this workaround does not work for all modular functions and in fact simply circumvents the modular operation. Explanation:
Consider $\hat{x_i} = s + i \cdot d \cdot \Phi$, which is the linear part with no modulo operation applied. Given the pertubations on these points we can perform linear regression and get $\hat{s} \approx s$ and $\hat{d} \approx d \cdot \hat{\Phi}$ with $\hat{\Phi}$ being the new ratio we're interested in. We then simply perform the modulo operation after the regression was done.
In essence in this case we can simply circumvent the modulo operation since it is the outermost operation in the right-hand-side syntax tree. I strongly suspect doing this is not possible for most other modular functions.