I have a linear model $$Y=HX+N,$$ where $H$ is a matrix and $X$ are drawn from $p_X(X)$, and $N$ is Gaussian noise variates.

Now, if $X$ is multivariate Gaussian, then a linear estimator $\hat{X}=A+BX$ solves the problem $$\mathcal{E}(|\hat{X}-X|^2).$$ The matrix $B$ is the standard Wiener filter, and $A$ is only activated if $\mathcal{E}(X)\neq 0$.

Now, for non-Gaussian inputs, a linear estimator is not optimal.

However, my intuition tells me that the solution could be expressed through a sum of cumulants. My reasoning is that since only the two first cumulants are non-zero for Gaussian, this is also the reason why a linear estimator $A+BX$ is optimal for Gaussian inputs $X$.

Is there anyone that have been thinking about this problem? (or is willing to do so?)