Noise properties of a linear system if elements are deleted

Dear All,

I'm hoping someone can help with a linear algebra problem. I have an
equation Cx = s where C is a matrix and s and x are vectors and i'm
trying to solve for x (obviously) all of which are complex valued. The
data in s are fairly noisy and the signal content is also sparse. By
this I mean that only a few components of x contain useful information
and the others are zero.

A simple inverse solution suffers from noise amplification because C is
usually not terribly well conditioned, and this is bad because s is
quite noisy. My solution is to use a model to guess which components of
x contain useful information and which are expected to be zero. I then
delete these elements of x and delete the corresponding columns in
matrix C. The inverse of this smaller matrix is better conditioned (if
my model was right) and generally this performs very well. My question
is that is my ad-hoc method an example of something more general and
well known? I am particularly interested in understanding the noise
propagation properties of this approach.

Cheers

Shaihan

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