Editing Least squares (section)

===Tikhonov regularization===
{{Main|Tikhonov regularization}}
In some contexts, a [[Regularization (machine learning)|regularized]] version of the least squares solution may be preferable. [[Tikhonov regularization]] (or [[ridge regression]]) adds a constraint that <math>\left\|\beta\right\|_2^2</math>, the squared [[L2-norm|<math>\ell_2</math>-norm]] of the parameter vector, is not greater than a given value to the least squares formulation, leading to a constrained minimization problem. This is equivalent to the unconstrained minimization problem where the objective function is the residual sum of squares plus a penalty term <math>\alpha \left\|\beta\right\|_2^2</math> and <math>\alpha</math> is a tuning parameter (this is the [[Lagrange multipliers|Lagrangian]] form of the constrained minimization problem).<ref>{{cite arXiv |last=van Wieringen |first=Wessel N. |year=2021 |title=Lecture notes on ridge regression |class=stat.ME |eprint=1509.09169 }}</ref>

In a [[Bayesian statistics|Bayesian]] context, this is equivalent to placing a zero-mean normally distributed [[prior distribution|prior]] on the parameter vector.