Generally, the answers $x$ of this least square problem $$Ax = 0$$ where $A = []_{m\times n}$ and $x = []_{n\times 1}$ are in the null space of $A$.
I know that people usually use the right-most vector as an approximate solution (a column vector that corresponds to the smallest singular value). However, if we recall another least square problem $||Ax - b||_2$, there is the technique called Tikhonov Regularization, i.e. solve the problem by using the following equation, $$(A^TA + \lambda R^T R)x = A^Tb$$ where $R$ is a regularization matrix and $\lambda$ is a positive real value. This equation is usually used instead of pseudoinverse when the problem is ill-posed (the solution $x$ is highly sensitive to the data in $A$ or $b$).
My question is that if I am trying to minimize $||Ax||_2$, is there any technique that is similar to Tikhonov regularization?
Thank you in advance for answers.