Editing Ridge regression (section)

==Lavrentyev regularization==
In some situations, one can avoid using the transpose <math>A^\mathsf{T}</math>, as proposed by [[Mikhail Lavrentyev]].<ref>{{cite book |first=M. M. |last=Lavrentiev |title=Some Improperly Posed Problems of Mathematical Physics |publisher=Springer |location=New York |year=1967 }}</ref> For example, if <math>A</math> is symmetric positive definite, i.e. <math>A = A^\mathsf{T} > 0</math>, so is its inverse <math>A^{-1}</math>, which can thus be used to set up the weighted norm squared <math>\left\|\mathbf x\right\|_P^2 = \mathbf x^\mathsf{T} A^{-1} \mathbf x</math> in the generalized Tikhonov regularization, leading to minimizing  
<math display="block">\left\|A \mathbf x - \mathbf b\right\|_{A^{-1}}^2 + \left\|\mathbf x - \mathbf x_0 \right\|_Q^2</math>
or, equivalently up to a constant term,
<math display="block">\mathbf x^\mathsf{T} \left(A+Q\right) \mathbf x - 2 \mathbf x^\mathsf{T} \left(\mathbf b + Q \mathbf x_0\right).</math>

This minimization problem has an optimal solution <math>\mathbf x^*</math> which can be written explicitly using the formula
<math display="block">\mathbf x^* = \left(A + Q\right)^{-1} \left(\mathbf b + Q \mathbf x_0\right),</math>
which is nothing but the solution of the generalized Tikhonov problem where <math>A = A^\mathsf{T} = P^{-1}.</math>

The Lavrentyev regularization, if applicable, is advantageous to the original Tikhonov regularization, since the Lavrentyev matrix <math>A + Q</math> can be better conditioned, i.e., have a smaller [[condition number]], compared to the Tikhonov matrix <math>A^\mathsf{T} A + \Gamma^\mathsf{T} \Gamma.</math>