Ridge regression

{{Short description|Regularization technique for ill-posed problems}} {{Regression bar}} '''Ridge regression''' (also known as '''Tikhonov regularization''', named for Andrey Tikhonov) is a method of estimating the coefficients of multiple-regression models in scenarios where the variables are highly correlated.<ref name=Hilt>{{cite book |last1=Hilt |first1=Donald E. |last2=Seegrist |first2=Donald W. |title=Ridge, a computer program for calculating ridge regression estimates |date=1977 |doi=10.5962/bhl.title.68934 |url=https://www.biodiversitylibrary.org/bibliography/68934 }}{{pn|date=April 2022}}</ref> It has been used in many fields including econometrics, chemistry, and engineering.<ref name=Gruber /> It is a method of regularization of ill-posed problems.{{efn|In statistics, the method is known as '''ridge regression''', in machine learning it and its modifications are known as '''weight decay''', and with multiple independent discoveries, it is also variously known as the '''Tikhonov–Miller method''', the '''Phillips–Twomey method''', the '''constrained linear inversion''' method, '''{{math|''L''₂}} regularization''', and the method of '''linear regularization'''. It is related to the Levenberg–Marquardt algorithm for non-linear least-squares problems.}} It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters.<ref>{{cite book |first=Peter |last=Kennedy |author-link=Peter Kennedy (economist) |title=A Guide to Econometrics |location=Cambridge |publisher=The MIT Press |edition=Fifth |year=2003 |isbn=0-262-61183-X |pages=205–206 |url=https://books.google.com/books?id=B8I5SP69e4kC&pg=PA205 }}</ref> In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff).<ref>{{cite book |first=Marvin |last=Gruber |title=Improving Efficiency by Shrinkage: The James–Stein and Ridge Regression Estimators |location=Boca Raton |publisher=CRC Press |year=1998 |pages=7–15 |isbn=0-8247-0156-9 |url=https://books.google.com/books?id=wmA_R3ZFrXYC&pg=PA7 }}</ref>

The theory was first introduced by Hoerl and Kennard in 1970 in their ''Technometrics'' papers "Ridge regressions: biased estimation of nonorthogonal problems" and "Ridge regressions: applications in nonorthogonal problems".<ref>{{cite journal |last1=Hoerl |first1=Arthur E. |last2=Kennard |first2=Robert W. |title=Ridge Regression: Biased Estimation for Nonorthogonal Problems |journal=Technometrics |date=1970 |volume=12 |issue=1 |pages=55–67 |doi=10.2307/1267351 |jstor=1267351 }}</ref><ref>{{cite journal |last1=Hoerl |first1=Arthur E. |last2=Kennard |first2=Robert W. |title=Ridge Regression: Applications to Nonorthogonal Problems |journal=Technometrics |date=1970 |volume=12 |issue=1 |pages=69–82 |doi=10.2307/1267352 |jstor=1267352 }}</ref><ref name=Hilt />

Ridge regression was developed as a possible solution to the imprecision of least square estimators when linear regression models have some multicollinear (highly correlated) independent variables—by creating a ridge regression estimator (RR). This provides a more precise ridge parameters estimate, as its variance and mean square estimator are often smaller than the least square estimators previously derived.<ref name=Jolliffe>{{cite book |last1=Jolliffe |first1=I. T. |title=Principal Component Analysis |date=2006 |publisher=Springer Science & Business Media |isbn=978-0-387-22440-4 |page=178 |url=https://books.google.com/books?id=6ZUMBwAAQBAJ&pg=PA178 }}</ref><ref name=Gruber>{{cite book |last1=Gruber |first1=Marvin |title=Improving Efficiency by Shrinkage: The James--Stein and Ridge Regression Estimators |date=1998 |publisher=CRC Press |isbn=978-0-8247-0156-7 |page=2 |url=https://books.google.com/books?id=wmA_R3ZFrXYC&pg=PA2 }}</ref>

==Overview== In the ordinary least squares solution of : <math> \mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\varepsilon}, \, </math> the problem of a near-singular moment matrix <math>\mathbf{X}^\mathsf{T}\mathbf{X}</math> is alleviated by adding positive elements to the diagonals, thereby decreasing its condition number. Compared to the ordinary least squares estimator, the simple ridge estimator has an extra term <math>\lambda \mathbf{I}</math> in the denominator: <math display="block">\hat{\boldsymbol{\beta}}_{\lambda} = \left(\mathbf{X}^{\mathsf{T}} \mathbf{X} + \lambda \mathbf{I}\right)^{-1} \mathbf{X}^{\mathsf{T}} \mathbf{Y}</math> where <math>\mathbf{Y}</math> is the regressand or response vector, <math>\mathbf{X}</math> is the design matrix, <math>\mathbf{I}</math> is the identity matrix, and the ridge (or Tikhonov) regularization parameter <math>\lambda \geq 0</math> serves as the constant shifting the diagonals of the moment matrix.<ref>For the choice of <math>\lambda</math> in practice, see {{cite journal |first1=Ghadban |last1=Khalaf |first2=Ghazi |last2=Shukur |title=Choosing Ridge Parameter for Regression Problems |journal=Communications in Statistics – Theory and Methods |volume=34 |year=2005 |issue=5 |pages=1177–1182 |doi=10.1081/STA-200056836 |s2cid=122983724 }}</ref> It can be shown that this estimator is the solution to the least squares problem subject to the constraint <math>\boldsymbol\beta^\mathsf{T}\boldsymbol\beta = c</math>, which can be expressed as a Lagrangian minimization: <math display="block"> \text{argmin}_{\boldsymbol{\beta}} \, \|\mathbf{Y} - \mathbf{X} \boldsymbol{\beta}\|^2 + \lambda \left(\boldsymbol\beta^\mathsf{T}\boldsymbol\beta - c\right)</math> which shows that <math>\lambda</math> is nothing but the Lagrange multiplier of the constraint.<ref>{{Cite arXiv|last=van Wieringen |first=Wessel |date=2021-05-31 |title=Lecture notes on ridge regression |class=stat.ME |eprint=1509.09169 }}</ref> In fact, there is a one-to-one relationship between <math>c</math> and <math>\lambda</math> and since, in practice, we do not know <math>c</math>, we define <math>\lambda</math> heuristically or find it via additional data-fitting strategies, see Determination of the Tikhonov parameter below.

Note that as <math>\lambda \downarrow 0</math>, the constraint eventually becomes non-binding, and the ridge estimator converges to the minimum-norm ordinary least squares estimator, here denoted as <math>\hat{\boldsymbol\beta}=\hat{\boldsymbol\beta}_0</math>:

<math display="block"> \lim_{\lambda\downarrow 0} \hat{\boldsymbol{\beta}}_{\lambda}=\mathbf{X}^+\mathbf{Y}=\hat{\boldsymbol\beta}_0, </math> with <math> \mathbf{X}^+</math> denoting the pseudoinverse of <math> \mathbf{X}</math>.

==Determination of the Tikhonov parameter== The optimal regularization parameter <math>\lambda</math> is usually unknown and in practice needs to be estimated. Typically, a data-driven choice for the Tikhonov regularization parameter <math>\lambda</math> is accomplished either via cross-validation, or via a plug-in procedure, as follows.

=== Generalized cross-validation estimator === A common data-driven choice for <math>\lambda</math> is the minimizer of the cross-validation loss or its generalizations. For example, Grace Wahba proved that the optimal parameter, in the sense of generalized cross-validation minimizes<ref>{{cite journal |last=Wahba |first=G. |year=1990 |title=Spline Models for Observational Data |journal=CBMS-NSF Regional Conference Series in Applied Mathematics |publisher=Society for Industrial and Applied Mathematics |bibcode=1990smod.conf.....W }}</ref><ref>{{cite journal |last3=Wahba |first3=G. |first1=G. |last1=Golub |first2=M. |last2=Heath |year=1979 |title=Generalized cross-validation as a method for choosing a good ridge parameter |journal=Technometrics |volume=21 |issue=2 |pages=215–223 |url=http://www.stat.wisc.edu/~wahba/ftp1/oldie/golub.heath.wahba.pdf |doi=10.1080/00401706.1979.10489751}}</ref>

<math display="block">G = \frac{\operatorname{RSS}}{\tau^2} = \frac{\left\|\mathbf{X} \hat{\boldsymbol\beta} - \mathbf Y\right\|^2}{ \left[\operatorname{tr}\left(\mathbf I - \mathbf{X} \left(\mathbf{X} ^\mathsf{T} \mathbf{X} + \lambda^2\mathbf I\right)^{-1} \mathbf{X} ^\mathsf{T}\right)\right]^2},</math> where <math>\operatorname{RSS}</math> is the residual sum of squares, and <math>\tau</math> is the effective number of degrees of freedom.

=== Plug-in estimator === Assume that <math> \mathbf{X} </math> is an <math>n\times p </math> matrix and define the matrix <math> \Omega:=(\mathbf{X}^\top\mathbf{X}/n)^+</math>. Then, consider the following choice for the Tikhonov regularization parameter:

<math display="block"> \lambda^*:=\frac{\varsigma^2\mathrm{tr}(\Omega)}{\boldsymbol\beta^\top\Omega\boldsymbol\beta+3\varsigma^2\mathrm{tr}(\Omega^2)/n}, </math>

where <math>\varsigma^2 </math> is the variance of the noise <math>\boldsymbol\varepsilon=\mathbf{Y}-\mathbf{X}\boldsymbol\beta</math>, that is, <math>\mathrm{Var}(\boldsymbol\varepsilon)=\varsigma^2 \mathbf I </math>. It can be shown<ref name=":0">{{Cite book |last=Botev |first=Zdravko I. |title=Data Science and Machine Learning: Mathematical and Statistical Methods |last2=Kroese |first2=Dirk P. |last3=Taimre |first3=Thomas |date= |publisher=CRC Press |year=2025 |isbn=978-1-032-48868-4 |edition=2nd |series= |location=Boca Raton ; London |publication-date=2025 |page=267-268 |language=English}}</ref> that the ridge estimator <math>\hat{\boldsymbol\beta}_{\lambda^*} </math> enjoys smaller expected in-sample risk than the minimum-norm least-squares estimator <math>\hat{\boldsymbol\beta}_0=\mathbf{X}^+\mathbf{Y}</math>. More precisely,

<math display="block"> \mathbb E\|\mathbf {Y}'-\mathbf{X}\hat{\boldsymbol\beta}_0\|^2\geq\mathbb E\|\mathbf{ Y}'-\mathbf{X}\hat{\boldsymbol\beta}_{\lambda^*}\|^2+\frac{\varsigma^2 }{n}\lambda^*\mathrm{tr}(\Omega), </math>

where the expectations treat <math>\mathbf{X} </math> as fixed and <math>\mathbf {Y}' </math> is ''test response'' data, independent from <math>\mathbf {Y} </math> (and hence independent from the estimators <math>\hat{\boldsymbol\beta}_0 </math> and <math>\hat{\boldsymbol\beta}_{\lambda^*} </math>).

Of course, in practice the formula for <math>\lambda^* </math> is used by plugging in statistical estimators for the unknown parameters <math>\boldsymbol\beta </math> and <math>\varsigma^2 </math>. When <math>n>p </math>, the most natural estimators for these parameters are the usual least-squares ones:

<math display="block"> \hat{\boldsymbol\beta}=\mathbf{X}^+\mathbf{Y},\qquad\hat\varsigma^2= \frac{\|\mathbf{Y}-\mathbf{X}\hat{\boldsymbol\beta}\|^2}{n-p}. </math> Replacing the unknown <math>\boldsymbol\beta, \varsigma^2 </math> in the formula for <math>\lambda^* </math> with the corresponding <math>\hat{\boldsymbol\beta}, \hat{\varsigma}^2 </math> thus gives the so-called ''plug-in estimator'' <math>\widehat\lambda^*</math> for the optimal <math>\lambda^* </math>.

Alternative approaches to the data-driven selection of the Tikhonov regularization parameter include the discrepancy principle, L-curve method,<ref>P. C. Hansen, "The L-curve and its use in the numerical treatment of inverse problems", [https://www.sintef.no/globalassets/project/evitameeting/2005/lcurve.pdf]</ref> restricted maximum likelihood.

==History== Tikhonov regularization was invented independently in many different contexts. It became widely known through its application to integral equations in the works of Andrey Tikhonov<ref>{{Cite journal| last=Tikhonov | first=Andrey Nikolayevich | author-link=Andrey Nikolayevich Tikhonov | year=1943 | title=Об устойчивости обратных задач |trans-title=On the stability of inverse problems | journal=Doklady Akademii Nauk SSSR | volume=39 | issue=5 | pages=195–198|url=http://a-server.math.nsc.ru/IPP/BASE_WORK/tihon_en.html| archive-url=https://web.archive.org/web/20050227163812/http://a-server.math.nsc.ru/IPP/BASE_WORK/tihon_en.html | archive-date=2005-02-27 }}</ref><ref>{{Cite journal| last=Tikhonov | first=A. N. | year=1963 | title=О решении некорректно поставленных задач и методе регуляризации | journal=Doklady Akademii Nauk SSSR | volume=151 | pages=501–504}}. Translated in {{Cite journal| journal=Soviet Mathematics | volume=4 | pages=1035–1038 | title=Solution of incorrectly formulated problems and the regularization method }}</ref><ref>{{Cite book| last=Tikhonov | first=A. N. |author2=V. Y. Arsenin | year=1977 | title=Solution of Ill-posed Problems | publisher=Winston & Sons | location=Washington | isbn=0-470-99124-0}}</ref><ref>{{cite book |last1=Tikhonov |first1=Andrey Nikolayevich |last2=Goncharsky |first2=A. |last3=Stepanov |first3=V. V. |last4=Yagola |first4=Anatolij Grigorevic |title=Numerical Methods for the Solution of Ill-Posed Problems |date=30 June 1995 |publisher=Springer Netherlands |location=Netherlands |isbn=0-7923-3583-X |url=https://www.springer.com/us/book/9780792335832 |access-date=9 August 2018 |ref=TikhonovSpringer1995Numerical}}</ref><ref>{{cite book |last1=Tikhonov |first1=Andrey Nikolaevich |last2=Leonov |first2=Aleksandr S. |last3=Yagola |first3=Anatolij Grigorevic |title=Nonlinear ill-posed problems |date=1998 |publisher=Chapman & Hall |location=London |isbn=0-412-78660-5 |url=https://www.springer.com/us/book/9789401751698 |access-date=9 August 2018 |ref=TikhonovChapmanHall1998Nonlinear}}</ref> and David L. Phillips.<ref>{{Cite journal | last1 = Phillips | first1 = D. L. | doi = 10.1145/321105.321114 | title = A Technique for the Numerical Solution of Certain Integral Equations of the First Kind | journal = Journal of the ACM | volume = 9 | pages = 84–97 | year = 1962 | s2cid = 35368397 | doi-access = free }}</ref> Some authors use the term '''Tikhonov–Phillips regularization'''. The finite-dimensional case was expounded by Arthur E. Hoerl, who took a statistical approach,<ref>{{cite journal |last1=Hoerl |first1=Arthur E. |title=Application of Ridge Analysis to Regression Problems |journal=Chemical Engineering Progress |date=1962 |volume=58 |issue=3 |pages=54–59 |ref=AEHoerl1962V58I3}}</ref> and by Manus Foster, who interpreted this method as a Wiener–Kolmogorov (Kriging) filter.<ref>{{Cite journal | last1 = Foster | first1 = M. | title = An Application of the Wiener-Kolmogorov Smoothing Theory to Matrix Inversion | doi = 10.1137/0109031 | journal = Journal of the Society for Industrial and Applied Mathematics | volume = 9 | issue = 3 | pages = 387–392 | year = 1961 }}</ref> Following Hoerl, it is known in the statistical literature as ridge regression,<ref>{{cite journal | last = Hoerl | first = A. E. |author2=R. W. Kennard | year = 1970 | title=Ridge regression: Biased estimation for nonorthogonal problems | journal=Technometrics | volume=12 | issue=1 | pages = 55–67 | doi=10.1080/00401706.1970.10488634}}</ref> named after ridge analysis ("ridge" refers to the path from the constrained maximum).<ref>{{Cite journal |last=Hoerl |first=Roger W. |date=2020-10-01 |title=Ridge Regression: A Historical Context |url=https://www.tandfonline.com/doi/full/10.1080/00401706.2020.1742207 |journal=Technometrics |language=en |volume=62 |issue=4 |pages=420–425 |doi=10.1080/00401706.2020.1742207 |issn=0040-1706|url-access=subscription }}</ref>

== Tikhonov regularization for linear equations== Suppose that for a known real matrix <math>A</math> and vector <math>\mathbf{b}</math>, we wish to find a vector <math>\mathbf{x}</math> such that <math display="block">A\mathbf{x} = \mathbf{b},</math> where <math>\mathbf{x}</math> and <math>\mathbf{b}</math> may be of different sizes and <math>A</math> may even be non-square.

The standard approach is ordinary least squares linear regression.{{Clarify|reason=does this represent a system of linear equations (i.e. are x and b both of the same dimension as one side of the - supposedly square - matrix? then, as far as I know, the standard approach for solving it is any of a wide range of solvers ''not'' including linear regression|date=May 2020}} However, if no <math>\mathbf{x}</math> satisfies the equation or more than one <math>\mathbf{x}</math> does—that is, the solution is not unique—the problem is said to be ill posed. In such cases, ordinary least squares estimation leads to an overdetermined, or more often an underdetermined system of equations. Most real-world phenomena have the effect of low-pass filters{{Clarify|reason=If multiplying a matrix by x is a filter, what in A is a frequency, and what values correspond to high or low frequencies?|date=November 2022}} in the forward direction where <math>A</math> maps <math>\mathbf{x}</math> to <math>\mathbf{b}</math>. Therefore, in solving the inverse-problem, the inverse mapping operates as a high-pass filter that has the undesirable tendency of amplifying noise (eigenvalues / singular values are largest in the reverse mapping where they were smallest in the forward mapping). In addition, ordinary least squares implicitly nullifies every element of the reconstructed version of <math>\mathbf{x}</math> that is in the null-space of <math>A</math>, rather than allowing for a model to be used as a prior for <math>\mathbf{x}</math>. Ordinary least squares seeks to minimize the sum of squared residuals, which can be compactly written as <math display="block">\left\|A\mathbf{x} - \mathbf{b}\right\|_2^2,</math> where <math>\|\cdot\|_2</math> is the Euclidean norm.

In order to give preference to a particular solution with desirable properties, a regularization term can be included in this minimization: <math display="block">\left\|A\mathbf{x} - \mathbf{b}\right\|_2^2 + \left\|\Gamma \mathbf{x}\right\|_2^2=\left\|\mathcal{A}\mathbf{x} - \mathcal{b}\right\|_2^2,</math> where <math>\mathcal{A}=\begin{pmatrix}A\\\Gamma\end{pmatrix}</math> and <math>\mathcal{b}=\begin{pmatrix}\mathbf{b}\\\boldsymbol0\end{pmatrix}</math>, for some suitably chosen '''Tikhonov matrix''' <math>\Gamma </math>. In many cases, this matrix is chosen as a scalar multiple of the identity matrix (<math>\Gamma = \alpha I</math>), giving preference to solutions with smaller norms; this is known as '''{{math|''L''₂}} regularization'''.<ref>{{cite conference |first=Andrew Y. |last=Ng |author-link=Andrew Ng |year=2004 |title=Feature selection, L1 vs. L2 regularization, and rotational invariance |conference=Proc. ICML |url=https://icml.cc/Conferences/2004/proceedings/papers/354.pdf}}</ref> In other cases, high-pass operators (e.g., a difference operator or a weighted Fourier operator) may be used to enforce smoothness if the underlying vector is believed to be mostly continuous. This regularization improves the conditioning of the problem, thus enabling a direct numerical solution. Treating it as an ordinary least squares problem with augmented matrices <math>\mathcal{A}</math> and <math>\mathcal{b}</math>, the solution is <math display="block">\hat\mathbf{x} =(\mathcal{A}^\mathsf{T} \mathcal{A})^{-1} \mathcal{A}^\mathsf{T} \mathbf{\mathcal{b}} = (A^\mathsf{T} A + \Gamma^\mathsf{T} \Gamma)^{-1} A^\mathsf{T} \mathbf{b}.</math> The effect of regularization may be varied by the scale of the matrix <math>\Gamma</math>. For <math>\Gamma = 0</math> this reduces to the unregularized least-squares solution, provided that (''A''^T''A'')⁻¹ exists. Note that in case of a complex matrix <math>A</math>, as usual the transpose <math>A^\mathsf{T}</math> has to be replaced by the Hermitian transpose <math>A^\mathsf{H}</math>.

{{math|''L''₂}} regularization is used in many contexts aside from linear regression, such as classification with logistic regression or support vector machines,<ref>{{cite journal |author1=R.-E. Fan |author2=K.-W. Chang |author3=C.-J. Hsieh |author4=X.-R. Wang |author5=C.-J. Lin |title=LIBLINEAR: A library for large linear classification |journal=Journal of Machine Learning Research |volume=9 |pages=1871–1874 |year=2008}}</ref> and matrix factorization.<ref>{{cite journal |last1=Guan |first1=Naiyang |first2=Dacheng |last2=Tao |first3=Zhigang |last3=Luo |first4=Bo |last4=Yuan |title=Online nonnegative matrix factorization with robust stochastic approximation |journal=IEEE Transactions on Neural Networks and Learning Systems |volume=23 |issue=7 |year=2012 |pages=1087–1099|doi=10.1109/TNNLS.2012.2197827 |pmid=24807135 |bibcode=2012ITNNL..23.1087G |s2cid=8755408 }}</ref>

=== Application to existing fit results ===

Since Tikhonov Regularization simply adds a quadratic term to the objective function in optimization problems, it is possible to do so after the unregularised optimisation has taken place. E.g., if the above problem with <math>\Gamma = 0</math> yields the solution <math>\hat\mathbf{x}_0</math>, the solution in the presence of <math>\Gamma \ne 0</math> can be expressed as: <math display="block">\hat\mathbf{x} = B \hat\mathbf{x}_0,</math> with the "regularisation matrix" <math>B = \left(A^\mathsf{T} A + \Gamma^\mathsf{T} \Gamma\right)^{-1} A^\mathsf{T} A</math>.

If the parameter fit comes with a covariance matrix of the estimated parameter uncertainties <math>V_0</math>, then the regularisation matrix will be <math display="block">B = (V_0^{-1} + \Gamma^\mathsf{T}\Gamma)^{-1} V_0^{-1},</math> and the regularised result will have a new covariance <math display="block">V = B V_0 B^\mathsf{T}.</math>

In the context of arbitrary likelihood fits, this is valid, as long as the quadratic approximation of the likelihood function is valid. This means that, as long as the perturbation from the unregularised result is small, one can regularise any result that is presented as a best fit point with a covariance matrix. No detailed knowledge of the underlying likelihood function is needed.<ref>{{cite journal|arxiv=2207.02125 |doi=10.1088/1748-0221/17/10/P10021 |title=Post-hoc regularisation of unfolded cross-section measurements |date=2022 |last1=Koch |first1=Lukas |journal=Journal of Instrumentation |volume=17 |issue=10 |article-number=10021 |bibcode=2022JInst..17P0021K }}</ref>

===Generalized Tikhonov regularization=== For general multivariate normal distributions for <math>\mathbf x</math> and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an <math>\mathbf x</math> to minimize <math display="block">\left\|A \mathbf x - \mathbf b\right\|_P^2 + \left\|\mathbf x - \mathbf x_0\right\|_Q^2,</math> where we have used <math>\left\|\mathbf{x}\right\|_Q^2</math> to stand for the weighted norm squared <math>\mathbf{x}^\mathsf{T} Q \mathbf{x}</math> (compare with the Mahalanobis distance). In the Bayesian interpretation <math>P</math> is the inverse covariance matrix of <math>\mathbf b</math>, <math>\mathbf x_0</math> is the expected value of <math>\mathbf x</math>, and <math>Q</math> is the inverse covariance matrix of <math>\mathbf x</math>.

The Tikhonov matrix is not explicitly included because the corresponding regularization term <math>\left\|\Gamma \mathbf x - \mathbf x_0'\right\|_{Q'}^2</math> reduces to above with <math>\Gamma \mathbf x_0=\mathbf x_0'</math> and <math>Q=\Gamma^T Q' \Gamma</math>. For normal regularization where <math>Q'=I</math>, the Tikhonov matrix then appears in the Cholesky factorization <math>Q = \Gamma^\mathsf{T} \Gamma</math> and is considered a whitening filter.

This generalized problem has an optimal solution <math>\hat\mathbf{x}</math> which can be written explicitly using the formula <math display="block">\mathbf \hat\mathbf{x} = \left(A^\mathsf{T} PA + Q\right)^{-1} \left(A^\mathsf{T} P \mathbf{b} + Q \mathbf{x}_0\right) = \mathbf x_0 + \left(A^\mathsf{T} P A + Q \right)^{-1} \left(A^\mathsf{T} P \left(\mathbf b - A \mathbf x_0\right)\right).</math>

==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>

==Regularization in Hilbert space== Typically discrete linear ill-conditioned problems result from discretization of integral equations, and one can formulate a Tikhonov regularization in the original infinite-dimensional context. In the above we can interpret <math>A</math> as a compact operator on Hilbert spaces, and <math>x</math> and <math>b</math> as elements in the domain and range of <math>A</math>. The operator <math>A^* A + \Gamma^\mathsf{T} \Gamma </math> is then a self-adjoint bounded invertible operator.

==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 of <math>A</math>.

==Relation to probabilistic formulation== The probabilistic formulation of an inverse problem introduces (when all uncertainties are Gaussian) a covariance matrix <math> C_M</math> representing the ''a priori'' uncertainties on the model parameters, and a covariance matrix <math> C_D</math> representing the uncertainties on the observed parameters.<ref>{{cite book |last1=Tarantola |first1=Albert |title=Inverse Problem Theory and Methods for Model Parameter Estimation |date=2005 |publisher=Society for Industrial and Applied Mathematics (SIAM) |location=Philadelphia |isbn=0-89871-792-2 |edition=1st |url=http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/SIAM/index.html |access-date=9 August 2018 |ref=ATarantolaSIAM2004}}</ref> In the special case when these two matrices are diagonal and isotropic, <math> C_M = \sigma_M^2 I </math> and <math> C_D = \sigma_D^2 I </math>, and, in this case, the equations of inverse theory reduce to the equations above, with <math> \alpha = {\sigma_D}/{\sigma_M} </math>.<ref>{{cite journal | last1 = Huang | first1 = Yunfei. | display-authors = etal | year = 2019 | title = Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells | journal = Scientific Reports | volume = 9 | number = 1| page = 537 | article-number = 539 | doi = 10.1038/s41598-018-36896-x | pmid = 30679578 | doi-access = free | pmc = 6345967 | arxiv = 1810.05848 | bibcode = 2019NatSR...9..539H }}</ref><ref>{{cite journal | last1 = Huang | first1 = Yunfei | last2 = Gompper | first2 = Gerhard | last3 = Sabass | first3 = Benedikt |year = 2020 | title = A Bayesian traction force microscopy method with automated denoising in a user-friendly software package | journal = Computer Physics Communications | volume = 256 | article-number = 107313 | doi = 10.1016/j.cpc.2020.107313 | arxiv = 2005.01377 | bibcode = 2020CoPhC.25607313H }}</ref>

==Bayesian interpretation== {{main|Bayesian interpretation of regularization}} {{Further|Minimum mean square error#Linear MMSE estimator for linear observation process}} Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix <math>\Gamma</math> seems rather arbitrary, the process can be justified from a Bayesian point of view.<ref>{{cite book |first1=Edward |last1=Greenberg |first2=Charles E. Jr. |last2=Webster |title=Advanced Econometrics: A Bridge to the Literature |location=New York |publisher=John Wiley & Sons |year=1983 |pages=207–213 |isbn=0-471-09077-8 }}</ref> Note that for an ill-posed problem one must necessarily introduce some additional assumptions in order to get a unique solution. Statistically, the prior probability distribution of <math>x</math> is sometimes taken to be a multivariate normal distribution.<ref>{{cite journal | last1 = Huang | first1 = Yunfei. | display-authors = etal | year = 2019 | title = Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells | journal = Scientific Reports | volume = 9 | number = 1| page = 537 | article-number = 539 | doi = 10.1038/s41598-018-36896-x | pmid = 30679578 | doi-access = free | pmc = 6345967 | arxiv = 1810.05848 | bibcode = 2019NatSR...9..539H }}</ref> For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation <math>\sigma _x</math>. The data are also subject to errors, and the errors in <math>b</math> are also assumed to be independent with zero mean and standard deviation <math>\sigma _b</math>. Under these assumptions the Tikhonov-regularized solution is the most probable solution given the data and the ''a priori'' distribution of <math>x</math>, according to Bayes' theorem.<ref>{{cite book |author=Vogel, Curtis R. |title=Computational methods for inverse problems |publisher=Society for Industrial and Applied Mathematics |location=Philadelphia |year=2002 |isbn=0-89871-550-4 }}</ref>

If the assumption of normality is replaced by assumptions of homoscedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the Gauss–Markov theorem entails that the solution is the minimal unbiased linear estimator.<ref>{{cite book |last=Amemiya |first=Takeshi |author-link=Takeshi Amemiya |year=1985 |title=Advanced Econometrics |publisher=Harvard University Press |pages=[https://archive.org/details/advancedeconomet00amem/page/60 60–61] |isbn=0-674-00560-0 |url-access=registration |url=https://archive.org/details/advancedeconomet00amem/page/60 }}</ref>

==See also== * LASSO estimator is another regularization method in statistics. * Elastic net regularization * Matrix regularization * L-curve

==Notes== {{notelist}}

==References== {{Reflist}}

==Further reading== *{{cite book |first=Marvin |last=Gruber |title=Improving Efficiency by Shrinkage: The James–Stein and Ridge Regression Estimators |location=Boca Raton |publisher=CRC Press |year=1998 |isbn=0-8247-0156-9 |url=https://books.google.com/books?id=wmA_R3ZFrXYC }} * {{cite book |last=Kress |first=Rainer |title=Numerical Analysis |location=New York |publisher=Springer |year=1998 |isbn=0-387-98408-9 |pages=86–90 |chapter=Tikhonov Regularization |chapter-url=https://books.google.com/books?id=Jv_ZBwAAQBAJ&pg=PA86 }} * {{Cite book | last1=Press | first1=W. H. | last2=Teukolsky | first2=S. A. | last3=Vetterling | first3=W. T. | last4=Flannery | first4=B. P. | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | location=New York | isbn=978-0-521-88068-8 | chapter=Section 19.5. Linear Regularization Methods | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1006}} * {{cite book |first1=A. K. Md. Ehsanes |last1=Saleh |first2=Mohammad |last2=Arashi |first3=B. M. Golam |last3=Kibria |title=Theory of Ridge Regression Estimation with Applications |location=New York |publisher=John Wiley & Sons |year=2019 |isbn=978-1-118-64461-4 |url=https://books.google.com/books?id=v0KCDwAAQBAJ }} * {{cite book |first=Matt |last=Taddy |title=Business Data Science: Combining Machine Learning and Economics to Optimize, Automate, and Accelerate Business Decisions |chapter=Regularization |pages=69–104 |location=New York |publisher=McGraw-Hill |year=2019 |isbn=978-1-260-45277-8 |chapter-url=https://books.google.com/books?id=yPOUDwAAQBAJ&pg=PA69 }}

{{Least_squares_and_regression_analysis}} {{Authority control}}

Category:Linear algebra Category:Estimation methods Category:Inverse problems Category:Regression analysis