LS Solution vs. Pre-Averaging: Which is More Effective for Noise Reduction?

[divB]

Hi,

I have a system of equations $\mathbf{y} = \mathbf{A}\mathbf{c}$ where the entries in $\mathbf{c}$ are small (say, K=10 elements) and the number equations (i.e., elements in $\mathbf{y}$) is huge (say, N=10000 elements).

I want to solve now for $\mathbf{c}$; this can be done using LS with the Pseudo inverse:

$$\mathbf{c} = \mathbf{A}^{\dagger} \mathbf{y}$$

However, the vector $\mathbf{y}$ is now heavily corrupted by noise (just assume iid Gaussian).

I could calculate the mean over M consecutive elements in $\mathbf{y}$ and rows in $\mathbf{A}$ in order to average over the noise. The system would be collapsed to a smaller system with N/M entries which would be solved via LS.

Now I ask the question: Is this better than directly using LS with the full system?

I doubt because that's the sense of LS. However, I was not able to "proof" this analytically.

Any help?
Thanks,

 

[chiro]

Hey divB.

Can you use the properties of a psuedo-inverse to show that this holds? (Recall that a pseudo-inverse has the property that C*C'*C = C)