I am given a number $N$ of 2D point pairs $(x_i,y_i)$ with corresponding values $(v_i,w_i)$. My task is now to find a linear matrix s.t.
$$ \begin{pmatrix} m_{x,1} & m_{y,1} & b_1 \\ m_{x,2} & m_{y,2} & b_2 \end{pmatrix} \cdot \begin{pmatrix}x_i\\y_i \\ 1\end{pmatrix} = \begin{pmatrix}v_i\\w_i\end{pmatrix} $$
for all $i=1,\cdots, N$. Of course, with a high number of $N$, this system is underdetermined.
Using the normal equation, I can know find the mean matrix, such that overall the Euclidian mean is optimized.
However, I have a lot of outliers in my pairs that disturb the Euclidian mean. Thus, I want to find the Euclidian median or something similar that is robust against outliers.
Any idea how I would do that?