Suppose I have a set of angles $\theta_k, 1 \leq k \leq n$, all between $[-\pi,\pi]$. Would the two minimization problems
$$ f_1(\theta) = \frac{1}{2n} \sum_{j}(\theta-\theta_j)^2 $$
and
$$ f_2(\theta) = \frac{1}{2n} \sum_j (1 - \cos(\theta-\theta_j))^2 $$
Lead to the same solution?
The closed form of the first one is the mean of all $\theta_j$'s, the second one I think it can only be solved with gradient descent or similar.
I don't think they're the same problem, since the second one can be transformed into the first one if all theta's are close to each other.
Is there formal way to prove/disprove they're the same problem?
Thank you
For $n=2$ they are equal. For $n=3$ they are not, since $\theta_1=\theta_2=0$, $\theta_3=\pi$ is a counterexample. We have $$f_2(\theta) = \frac{1}{6} \left(2\left(1-\cos(\theta)\right)^2 + \left(1-\cos(\theta-\pi)\right)^2\right),$$ whose minimum does not occur at $\pi/3 \approx 1.05$ as can be seen in this graph.