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Ridge regression
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==Relation to singular-value decomposition and Wiener filter== With <math>\Gamma = \alpha I</math>, this least-squares solution can be analyzed in a special way using the [[singular-value decomposition]]. Given the singular value decomposition <math display="block">A = U \Sigma V^\mathsf{T}</math> with singular values <math>\sigma _i</math>, the Tikhonov regularized solution can be expressed as <math display="block">\hat{x} = V D U^\mathsf{T} b,</math> where <math>D</math> has diagonal values <math display="block">D_{ii} = \frac{\sigma_i}{\sigma_i^2 + \alpha^2}</math> and is zero elsewhere. This demonstrates the effect of the Tikhonov parameter on the [[condition number]] of the regularized problem. For the generalized case, a similar representation can be derived using a [[generalized singular-value decomposition]].<ref name="Hansen_SIAM_1998">{{cite book |last1=Hansen |first1=Per Christian |title=Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion |date=Jan 1, 1998 |publisher=SIAM |location=Philadelphia, USA |isbn=978-0-89871-403-6 |edition=1st }}</ref> Finally, it is related to the [[Wiener filter]]: <math display="block">\hat{x} = \sum _{i=1}^q f_i \frac{u_i^\mathsf{T} b}{\sigma_i} v_i,</math> where the Wiener weights are <math>f_i = \frac{\sigma _i^2}{\sigma_i^2 + \alpha^2}</math> and <math>q</math> is the [[Rank (linear algebra)|rank]] of <math>A</math>.
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