Recovering angles from noisy matrix I have a matrix $A$ whose components are given by:
$$A_{ij}=\sin(\theta_i-\theta_j)+N_{ij}$$
where $N_{ij}$ is some noise - you're free to assume $N_{ij}=-N_{ji} \sim N(0,\sigma^2)$ and independent. My goal is to recover the angles $\theta_i$, which are fixed (assuming e.g. $\theta_1=0$ as a reference). I imagine there's some easy way of doing this (some sort of transform), but I don't personally know how, so any insight would be appreciated.
By the way, this is for a personal project. 
 A: Assume that $i,j$ runs from $1$ to $n$ and assume for a while that there is no noise. Apllying the $\arcsin(\cdot)$ function and recalling $\theta_1=0$ we obtain $\frac{n(n-1)}{2}$ linear equations in $n-1$ variables, and we can efficiently solve these equations. However, the noise prevents us from a straightforward use of the $\arcsin(\cdot)$ function since it is defined only on $[-1,1]$. We could do some (probably nonlinear) averaging if there were multiple measurements.
Update. Suppose that for the fixed value $\theta_i$ we have a set of measurements of $\mu_i=\sin(\theta_i)$ with an additive noise $\mathcal{N}(0,\sigma^2)$. Next we apply the projection $\mu_i^*=\min(1,\max(-1, \mu_i))$. Then the expected value of $\theta_i^*=\arcsin(\mu_i^*)$ is $\theta_i$. Moreover, since $\mu_i$ and $\mu_j$ are independednt, the $\theta^*_i$ and $\theta^*_j$ are also independednt. Thus, if we further construct a system of linear equalities and apply the least squares approach, we may expect the unbiased and effective estimate.
A: We should try to focus the problem. There are multiple ways to interpret this problem. What is your model? What are your measurements? What are we trying to solve?
Assume $\mathbf{A}\in\mathbb{C}^{m\times n}$. Assume the entry in row $r$ and column $c$ represents a measurement of $\mathbf{A}_{r,c}$.
Don't try to measure two angles. Just compute the difference 
$$
 \delta = \theta_{1} - \theta_{2}
$$
1 Column vectors are constant
$$\mathbf{A}_{r,c} = \delta_{c}$$
$\delta_{j}$ is constant for each column $c$. The least squares solution for column $r$ is the average of the column entries:
$$
 \delta_{c} = \arcsin \frac{1}{m} \sum_{k=1}^{m} \mathbf{A}_{r,k}
 = \arcsin \frac{1}{m} \sum_{k=1}^{m} \sin \delta_{r,k}
$$ 
2 Each matrix element represents a different angle
There is no solution. In essence you have $\gamma$ and are trying to solve for $\alpha$ and $\beta$ in
$$
 \sin ( \alpha - \beta ) = \gamma
$$
