How can one best estimate points coordinates, given their error-prone pair-wise distance?

Question

Let $P$ be a set of $n+t$ ($t > 0$) points in $\mathbb{R}^n$ and $M = (m_{i,j})_{i,j=1, \dots, n}$ be the matrix containing all measured pairwise euclidean distances between point $p_i$ and point $p_j$ with an additional normally distributed error $e \sim \mathcal{N}(0, \vartheta)$.

What is the best estimate of the coordinates of the points?

My idea

Without loss of generality, $p_1 = (0, \dots, 0)$ and $p_2 = (0, m_{1,2}, 0, \dots, 0)$. $p_3$ then is on the hypersphere around $p_1$ with distance $m_{1,3}$ and on the hypersphere around $p_2$ with the distance $m_{2,3}$. I could calculate a position of the next point with circle-circle intersection, but then I have the problem that I didn't take care of the errors

Answer 1

I will break down the problem to a problem in which we have a point $p$ and we have measured the distances $d_i$ to $p$ from the points $q_i$ for $i=1,..., N$.

then we can formulate the equations

$$d^2_1=[p-q_1]^T[p-q_1]+\varepsilon_1$$ $$d^2_2=[p-q_2]^T[p-q_2]+\varepsilon_2$$ $$\vdots $$ $$d^2_N=[p-q_N]^T[p-q_N]+\varepsilon_N$$

Then we can try to estimate the position $p$ by using the least squares estimate $\hat{p}$ by minimizing the cost function $F$ (this is equivalent to minimizing the sum of squared errors)

$$F(p,q_1,q_2,...,q_N)=\sum_{i=1}^{N}\left[d^2_N-[p-q_i]^T[p-q_i] \right]^2.$$

You do this for all your points then you update the distance matrix and try to run the same algorithm again to obtain a better estimate. You can also directly update the distances after calculating the first point and so forth to improve the convergence of the algorithm.