Consider a scalar system with $2K$ outputs and $K+2$ unknowns: $y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$.
The variables $n_{k,\ell}$ are zero mean noise variables. To estimate $a_1$ and $a_2$, we can, e.g., use the generalized method of moments with moment condition: $$\mathbb{E}\left(y_{k,1}a_2-y_{k,2}a_1\right)=0.$$ The expectation here is over the noise only. The unknown $x_k$ is assumed deterministic but unknown. In other words, I can estimate them by solving the quadratic optimization $$\min_{a_1,a_2}\sum_{k} |y_{k,1}a_2-y_{k,2}a_1|^2.$$
Now, move on to a vector case. Each pair of signals (one $k$) would now read:$y_1 = Xa_1 +n_1 \quad y_1 = Xa_2 +n_2,$ where $X$ is an $N\times N$ unknown matrix and all other quantities are $N\times 1$ vectors.
In order to apply the same technique as in the scalar case, and thereby obtaining a quadratic problem, I need to find mappings $f_{a_1}(\cdot)$ and $f_{a_2}(\cdot)$ that are linear in $a_1$ and $a_2$ such that: $$\mathbb{E}\left(f_{a_1}(y_1,y_2)-f_{a_2}(y_1,y_2)\right) = 0.$$
I have tried for quite some time now, but cannot come up with anything.
If the mapping is not linear, the resulting problem is not quadratic and is not easy to solve as I have a huge system (many $k$'s).
So, are there such mappings?
