A question about ICA and non-gaussianity

Question

I'm new in the ICA processing and I'm trying to understand the non-gaussian requirement. I read that the problem is that, if the composed data is $\mathbf{x}=\mathbf{As}$ with $\mathbf{A}$ (unknown) mixing matrix and $\mathbf{s}$ (unknown) original signals, then in ICA framework the components of $\mathbf{s}$ can not be considered gaussian because no "pattern" can be found in $\mathbf{x}$ to recover a unique $\mathbf{A}$ (since any rotation is good). So my questions are:

1. (maybe a naive or stupid question, but I need to do to clarify my ideas): it seems that the problem is not that there is no solution, but that there are too many possible solutions. Why this should be a problem? If we see the problem as to estimate an $\mathbf{A}$ s.t. $\mathbf{x}=\mathbf{As}$, we simply can tell that there are many solutions and return just one of them.. Where is my error?
2. The central Limit Theorem states that the sum of many independent signals goes toward the gaussian distribution, so ICA searches for the best non-gaussian $\mathbf{s}$. But, if the sum of many signals is gaussian, again we have that no "pattern" can be found in $\mathbf{x}$ to recover a unique $\mathbf{A}$ (again, any rotation is good), so we have the gaussian problem again. Where I wrong?

I apologize for the possibly stupid questions, but I think there is something that is not clear to me.

Answer 1

1. The problem with the first point is that if we cannot restrict the set of possible answers, then we will not be able to identify the ground truth components. So we will have a solution that is as useful as the others, but we cannot guarantee that what we found are the components that generated the data. E.g. if you have the mix of two people talking, you want to recover both audio signals although you probably can generate the mixed signal in many ways, but you are interested what each person said.

2. When your signal is Gaussian (either because it is Gaussian or due to the CLT), you will not be able to identify it. ICA makes the assumption that maximum one source can be Gaussian (with one component you can avoid rotational symmetry).

Reasons for the non-Gaussianity assumption

You want to predict the data distribution $p_x = A p_s.$ Usually, it is done by maximizing the (log)likelihood (denoting the inverse of the mixing matrix as $W$, and using the change of variable formula for probabilities) $$ \max_W \log p_x = \max_W \log [p_s *|\det W|] = \max_W \log p_s + \log |\det W| $$ If the sources are Gaussian, then for any rotation $R : Rp_s = p_s, $ and as rotations have $\det R = 1$, the loss function will be the same, namely: $$ \max_W \log [p_s *|\det RW|] = \max_W \log [p_s *|\det W||\det R|] = \max_W \log [p_s *|\det W|] $$