ICA for Noise Reduction over a Dataset
This appears similar to a classic least squared solution of an overdetermined equation that proceeds as follows:

Starting with:

$$ \mathbf{x} = \mathbf{A} \mathbf{s}  $$

$\mathbf{A}$ is not a square matrix if overdetermined (more equations than unknownns) so therefore an inverse does not exist. What you do then is multiply both sides by the transpose of $\mathbf{A}$ since $\mathbf{A^TA}$ is a square matrix. From that, assuming and inverse exists for$\mathbf{A^TA}$, you can solve for s:

$$ \mathbf{A^Tx} = \mathbf{A^TA} \mathbf{s}  $$

$$ \mathbf{s} =  (\mathbf{A^TA})^{-1}\mathbf{A^Tx} $$