Suppose my dataset consists of $N$ example vectors $\mathbf{x}_{1}, \ldots, \mathbf{x}_{N}$ where $\mathbf{x}_{n} \in \mathbb{R}^{p}$ $\forall n$. I assume that each vector $\mathbf{x}_{n}$ is comprised of an underlying true datapoint $\mathbf{s}_{n}$ that is corrupted by additive Gaussian noise $\mathbf{x}_{n} = \mathbf{s}_{n} + \mathbf{w}_{n}$.

What I want to do is estimate the $\mathbf{s}_{n}$. Can I do this with ICA?

So the ICA model is of the form:

$$ \mathbf{x}_{n} = a_{1,n} \mathbf{s}_{1} + a_{2,n} \mathbf{s}_{2} + \ldots + a_{N,n} \mathbf{s}_{N} $$

Does this mean I need to fit my problem to the model by assuming that $ \mathbf{s}_{n} = a_{1,n} \mathbf{s}_{1} $ and $\mathbf{w}_{n} = a_{2,n} \mathbf{s}_{2}$ ? If so, how do I recover the vectors $\mathbf{s}_{n}$ $\forall n$?

I ask because I see the ICA problem formulated as:

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

Would this then imply that: $\mathbf{s} = \mathbf{A}^{+} \mathbf{x}$, where $\mathbf{A}^{+}$ is the psuedo-inverse of $\mathbf{A}$? But how would I get the $\mathbf{s}_{n}$ out of this formulation?


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