Consider linear regression with some regularization: E.g. Find $x$ that minimizes $||Ax - b||^2+\lambda||x||_1$
Usually, columns of A are standardized to have zero mean and unit norm, while $b$ is centered to have zero mean. I want to make sure if my understanding of the reason for standardizing and centering is correct.
By making the means of columns of $A$ and $b$ zero, we don't need an intercept term anymore. Otherwise, the objective would have been $||Ax-x_01-b||^2+\lambda||x||_1$. By making the norms of columns of A equal to 1, we remove the possibility of a case where just because one column of A has very high norm, it gets a low coefficient in $x$, which might lead us to conclude incorrectly that that column of A doesn't "explain" $x$ well.
This reasoning is not exactly rigorous but intuitively, is that the right way to think?