I'm trying to solve the following least squares problem:

$\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$

where $Ax = b$ and $\tilde{b} = b + w$

### Question:

How do I determine which probability distribution fits $w$ best?

Also, $A \in \mathbb{R}^{n\times n}$ is a large, and sparse Toeplitz matrix, $\tilde{b} \in \mathbb{R}^{n \times 1}$

$w$ is not your typical measurement noise and is probably not Gaussian. Also, I have access to $b$ (training data) and therefore, $w$.

Also, in a related question, I'm using LSQR to determine $x$ --- since I have my training data, is there any way to do cross-validation to determine how to best solve for $x$?

For example, I tried Tikhonov regularization to find the best $\lambda$ but I'm looking for a better method.