# ICA for Noise Reduction over a Dataset

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?