Ridge regression

 ( Linear Algebra ) 

Ridge regression (also known as Tikhonov regularization, named for Andrey Tikhonov) is a method of estimating the of multiple- in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. It is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters.

In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff).

Marvin Gruber (1998). 9780824701567, CRC Press. . ISBN 9780824701567

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".

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.

(1998). 9780824701567, CRC Press. . ISBN 9780824701567

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Overview In the simplest case, the problem of a near-singular moment matrix $\backslash mathbf\{X\}^\backslash mathsf\{T\}\backslash mathbf\{X\}$ is alleviated by adding positive elements to the Main diagonal, thereby decreasing its condition number. Analogous to the ordinary least squares estimator, the simple ridge estimator is then given by $$\backslash hat\{\backslash boldsymbol\{\backslash beta\}\}\_\{R\}\; =\; \backslash left(\backslash mathbf\{X\}^\{\backslash mathsf\{T\}\}\; \backslash mathbf\{X\}\; +\; \backslash lambda\; \backslash mathbf\{I\}\backslash right)^\{-1\}\; \backslash mathbf\{X\}^\{\backslash mathsf\{T\}\}\; \backslash mathbf\{y\}$$ where $\backslash mathbf\{y\}$ is the regressand, $\backslash mathbf\{X\}$ is the design matrix, $\backslash mathbf\{I\}$ is the identity matrix, and the ridge parameter $\backslash lambda\; \backslash geq\; 0$ serves as the constant shifting the diagonals of the moment matrix.For the choice of $\backslash lambda$ in practice, see It can be shown that this estimator is the solution to the least squares problem subject to the constraint $\backslash boldsymbol\backslash beta^\backslash mathsf\{T\}\backslash boldsymbol\backslash beta\; =\; c$, which can be expressed as a Lagrangian minimization: $$\backslash hat\{\backslash boldsymbol\backslash beta\}\_\{R\}\; =\; \backslash text\{argmin\}\_\{\backslash boldsymbol\{\backslash beta\}\}\; \backslash ,\; \backslash left(\backslash mathbf\{y\}\; -\; \backslash mathbf\{X\}\; \backslash boldsymbol\{\backslash beta\}\backslash right)^\backslash mathsf\{T\}\; \backslash left(\backslash mathbf\{y\}\; -\; \backslash mathbf\{X\}\; \backslash boldsymbol\backslash beta\backslash right)\; +\; \backslash lambda\; \backslash left(\backslash boldsymbol\backslash beta^\backslash mathsf\{T\}\backslash boldsymbol\backslash beta\; -\; c\backslash right)$$ which shows that $\backslash lambda$ is nothing but the Lagrange multiplier of the constraint. In fact, there is a one-to-one relationship between $c$ and $\backslash beta$ and since, in practice, we do not know $c$, we define $\backslash lambda$ heuristically or find it via additional data-fitting strategies, see Determination of the Tikhonov factor.

Note that, when $\backslash lambda\; =\; 0$, in which case the constraint is non-binding, the ridge estimator reduces to ordinary least squares. A more general approach to Tikhonov regularization is discussed below.

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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. Translated in

A. N. Tikhonov (1977). 9780470991244, Winston & Sons. ISBN 9780470991244

(1995). 079233583X, Springer Netherlands. . 079233583X

(1998). 9780412786600, Chapman & Hall. . ISBN 9780412786600

and David L. Phillips. Some authors use the term Tikhonov–Phillips regularization. The finite-dimensional case was expounded by Arthur E. Hoerl, who took a statistical approach, and by Manus Foster, who interpreted this method as a Kriging filter. Following Hoerl, it is known in the statistical literature as ridge regression, named after ridge analysis ("ridge" refers to the path from the constrained maximum).

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Tikhonov regularization Suppose that for a known real matrix $A$ and vector $\backslash mathbf\{b\}$, we wish to find a vector $\backslash mathbf\{x\}$ such that $$A\backslash mathbf\{x\}\; =\; \backslash mathbf\{b\},$$ where $\backslash mathbf\{x\}$ and $\backslash mathbf\{b\}$ may be of different sizes and $A$ may be non-square.

The standard approach is ordinary least squares linear regression. However, if no $\backslash mathbf\{x\}$ satisfies the equation or more than one $\backslash mathbf\{x\}$ 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 in the forward direction where $A$ maps $\backslash mathbf\{x\}$ to $\backslash mathbf\{b\}$. 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 $\backslash mathbf\{x\}$ that is in the null-space of $A$, rather than allowing for a model to be used as a prior for $\backslash mathbf\{x\}$. Ordinary least squares seeks to minimize the sum of squared residuals, which can be compactly written as $$\backslash left\backslash |A\backslash mathbf\{x\}\; -\; \backslash mathbf\{b\}\backslash right\backslash |\_2^2,$$ where $\backslash |\backslash cdot\backslash |\_2$ 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: $$\backslash left\backslash |A\backslash mathbf\{x\}\; -\; \backslash mathbf\{b\}\backslash right\backslash |\_2^2\; +\; \backslash left\backslash |\backslash Gamma\; \backslash mathbf\{x\}\backslash right\backslash |\_2^2=\backslash left\backslash |\backslash begin\{pmatrix\}A\backslash \backslash \backslash Gamma\backslash end\{pmatrix\}\backslash mathbf\{x\}\; -\; \backslash begin\{pmatrix\}\backslash mathbf\{b\}\backslash \backslash \backslash boldsymbol0\backslash end\{pmatrix\}\backslash right\backslash |\_2^2$$ for some suitably chosen Tikhonov matrix $\backslash Gamma$. In many cases, this matrix is chosen as a scalar multiple of the identity matrix ($\backslash Gamma\; =\; \backslash alpha\; I$), giving preference to solutions with smaller norms; this is known as regularization. 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. An explicit solution, denoted by $\backslash hat\backslash mathbf\{x\}$, is given by $$\backslash hat\backslash mathbf\{x\}\; =\; \backslash left(A^\backslash mathsf\{T\}\; A\; +\; \backslash Gamma^\backslash mathsf\{T\}\; \backslash Gamma\backslash right)^\{-1\}\; A^\backslash mathsf\{T\}\; \backslash mathbf\{b\}=\; \backslash left(\backslash begin\{pmatrix\}A\backslash \backslash \backslash Gamma\backslash end\{pmatrix\}^\backslash mathsf\{T\}\backslash begin\{pmatrix\}A\backslash \backslash \backslash Gamma\backslash end\{pmatrix\}\backslash right)^\{-1\}\; \backslash begin\{pmatrix\}A\backslash \backslash \backslash Gamma\backslash end\{pmatrix\}^\backslash mathsf\{T\}\; \backslash begin\{pmatrix\}\backslash mathbf\{b\}\backslash \backslash \backslash boldsymbol0\backslash end\{pmatrix\}.$$ The effect of regularization may be varied by the scale of matrix $\backslash Gamma$. For $\backslash Gamma\; =\; 0$ this reduces to the unregularized least-squares solution, provided that ( A^T A)⁻¹ exists. Note that in case of a complex matrix $A$, as usual the transpose $A^\backslash mathsf\{T\}$ has to be replaced by the Hermitian transpose $A^\backslash mathsf\{H\}$.

regularization is used in many contexts aside from linear regression, such as classification with logistic regression or support vector machines, and matrix factorization.
     

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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 $\backslash Gamma\; =\; 0$ yields the solution $\backslash hat\backslash mathbf\{x\}\_0$, the solution in the presence of $\backslash Gamma\; \backslash ne\; 0$ can be expressed as: $$\backslash hat\backslash mathbf\{x\}\; =\; B\; \backslash hat\backslash mathbf\{x\}\_0,$$ with the "regularisation matrix" $B\; =\; \backslash left(A^\backslash mathsf\{T\}\; A\; +\; \backslash Gamma^\backslash mathsf\{T\}\; \backslash Gamma\backslash right)^\{-1\}\; A^\backslash mathsf\{T\}\; A$.

If the parameter fit comes with a covariance matrix of the estimated parameter uncertainties $V\_0$, then the regularisation matrix will be $$B\; =\; (V\_0^\{-1\}\; +\; \backslash Gamma^\backslash mathsf\{T\}\backslash Gamma)^\{-1\}\; V\_0^\{-1\},$$ and the regularised result will have a new covariance $$V\; =\; B\; V\_0\; B^\backslash mathsf\{T\}.$$

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.

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Generalized Tikhonov regularization For general multivariate normal distributions for $\backslash mathbf\; x$ and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an $\backslash mathbf\; x$ to minimize $$\backslash left\backslash |A\; \backslash mathbf\; x\; -\; \backslash mathbf\; b\backslash right\backslash |\_P^2\; +\; \backslash left\backslash |\backslash mathbf\; x\; -\; \backslash mathbf\; x\_0\backslash right\backslash |\_Q^2,$$ where we have used $\backslash left\backslash |\backslash mathbf\{x\}\backslash right\backslash |\_Q^2$ to stand for the weighted norm squared $\backslash mathbf\{x\}^\backslash mathsf\{T\}\; Q\; \backslash mathbf\{x\}$ (compare with the Mahalanobis distance). In the Bayesian interpretation $P$ is the inverse covariance matrix of $\backslash mathbf\; b$, $\backslash mathbf\; x\_0$ is the expected value of $\backslash mathbf\; x$, and $Q$ is the inverse covariance matrix of $\backslash mathbf\; x$. The Tikhonov matrix is then given as a factorization of the matrix $Q\; =\; \backslash Gamma^\backslash mathsf\{T\}\; \backslash Gamma$ (e.g. the Cholesky factorization) and is considered a whitening filter.

This generalized problem has an optimal solution $\backslash mathbf\; x^*$ which can be written explicitly using the formula $$\backslash mathbf\; x^*\; =\; \backslash left(A^\backslash mathsf\{T\}\; PA\; +\; Q\backslash right)^\{-1\}\; \backslash left(A^\backslash mathsf\{T\}\; P\; \backslash mathbf\{b\}\; +\; Q\; \backslash mathbf\{x\}\_0\backslash right),$$ or equivalently, when Q is not a null matrix: $$\backslash mathbf\; x^*\; =\; \backslash mathbf\; x\_0\; +\; \backslash left(A^\backslash mathsf\{T\}\; P\; A\; +\; Q\; \backslash right)^\{-1\}\; \backslash left(A^\backslash mathsf\{T\}\; P\; \backslash left(\backslash mathbf\; b\; -\; A\; \backslash mathbf\; x\_0\backslash right)\backslash right).$$

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Lavrentyev regularization In some situations, one can avoid using the transpose $A^\backslash mathsf\{T\}$, as proposed by Mikhail Lavrentyev. For example, if $A$ is symmetric positive definite, i.e. $A\; =\; A^\backslash mathsf\{T\}\; >\; 0$, so is its inverse $A^\{-1\}$, which can thus be used to set up the weighted norm squared $\backslash left\backslash |\backslash mathbf\; x\backslash right\backslash |\_P^2\; =\; \backslash mathbf\; x^\backslash mathsf\{T\}\; A^\{-1\}\; \backslash mathbf\; x$ in the generalized Tikhonov regularization, leading to minimizing $$\backslash left\backslash |A\; \backslash mathbf\; x\; -\; \backslash mathbf\; b\backslash right\backslash |\_\{A^\{-1\}\}^2\; +\; \backslash left\backslash |\backslash mathbf\; x\; -\; \backslash mathbf\; x\_0\; \backslash right\backslash |\_Q^2$$ or, equivalently up to a constant term, $$\backslash mathbf\; x^\backslash mathsf\{T\}\; \backslash left(A+Q\backslash right)\; \backslash mathbf\; x\; -\; 2\; \backslash mathbf\; x^\backslash mathsf\{T\}\; \backslash left(\backslash mathbf\; b\; +\; Q\; \backslash mathbf\; x\_0\backslash right).$$

This minimization problem has an optimal solution $\backslash mathbf\; x^*$ which can be written explicitly using the formula $$\backslash mathbf\; x^*\; =\; \backslash left(A\; +\; Q\backslash right)^\{-1\}\; \backslash left(\backslash mathbf\; b\; +\; Q\; \backslash mathbf\; x\_0\backslash right),$$ which is nothing but the solution of the generalized Tikhonov problem where $A\; =\; A^\backslash mathsf\{T\}\; =\; P^\{-1\}.$

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

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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 $A$ as a compact operator on , and $x$ and $b$ as elements in the domain and range of $A$. The operator $A^*\; A\; +\; \backslash Gamma^\backslash mathsf\{T\}\; \backslash Gamma$ is then a self-adjoint bounded invertible operator.

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Relation to singular-value decomposition and Wiener filter With $\backslash Gamma\; =\; \backslash alpha\; I$, this least-squares solution can be analyzed in a special way using the singular-value decomposition. Given the singular value decomposition $$A\; =\; U\; \backslash Sigma\; V^\backslash mathsf\{T\}$$ with singular values $\backslash sigma\; \_i$, the Tikhonov regularized solution can be expressed as $$\backslash hat\{x\}\; =\; V\; D\; U^\backslash mathsf\{T\}\; b,$$ where $D$ has diagonal values $$D\_\{ii\}\; =\; \backslash frac\{\backslash sigma\_i\}\{\backslash sigma\_i^2\; +\; \backslash alpha^2\}$$ 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.

(1998). 9780898714036, SIAM. ISBN 9780898714036

Finally, it is related to the Wiener filter: $$\backslash hat\{x\}\; =\; \backslash sum\; \_\{i=1\}^q\; f\_i\; \backslash frac\{u\_i^\backslash mathsf\{T\}\; b\}\{\backslash sigma\_i\}\; v\_i,$$ where the Wiener weights are $f\_i\; =\; \backslash frac\{\backslash sigma\; \_i^2\}\{\backslash sigma\_i^2\; +\; \backslash alpha^2\}$ and $q$ is the rank of $A$.

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Determination of the Tikhonov factor The optimal regularization parameter $\backslash alpha$ is usually unknown and often in practical problems is determined by an ad hoc method. A possible approach relies on the Bayesian interpretation described below. Other approaches include the discrepancy principle, cross-validation, L-curve method,P. C. Hansen, "The L-curve and its use in the numerical treatment of inverse problems", [1] restricted maximum likelihood and unbiased predictive risk estimator. Grace Wahba proved that the optimal parameter, in the sense of leave-one-out cross-validation minimizes $$G\; =\; \backslash frac\{\backslash operatorname\{RSS\}\}\{\backslash tau^2\}\; =\; \backslash frac\{\backslash left\backslash |X\; \backslash hat\{\backslash beta\}\; -\; y\backslash right\backslash |^2\}\{\; \backslash left\backslash operatorname\{Tr\}\backslash left(I^2\},$$ where $\backslash operatorname\{RSS\}$ is the residual sum of squares, and $\backslash tau$ is the effective number of degrees of freedom.

Using the previous SVD decomposition, we can simplify the above expression: $$\backslash operatorname\{RSS\}\; =\; \backslash left\backslash |\; y\; -\; \backslash sum\_\{i=1\}^q\; (u\_i\text{'}\; b)\; u\_i\; \backslash right\backslash |^2\; +\; \backslash left\backslash |\; \backslash sum\; \_\{i=1\}^q\; \backslash frac\{\backslash alpha^2\}\{\backslash sigma\_i^2\; +\; \backslash alpha^2\}\; (u\_i\text{'}\; b)\; u\_i\; \backslash right\backslash |^2,$$ $$\backslash operatorname\{RSS\}\; =\; \backslash operatorname\{RSS\}\_0\; +\; \backslash left\backslash |\; \backslash sum\_\{i=1\}^q\; \backslash frac\{\backslash alpha^2\}\{\backslash sigma\_i^2\; +\; \backslash alpha^2\}\; (u\_i\text{'}\; b)\; u\_i\; \backslash right\backslash |^2,$$ and $$\backslash tau\; =\; m\; -\; \backslash sum\_\{i=1\}^q\; \backslash frac\{\backslash sigma\_i^2\}\{\backslash sigma\_i^2\; +\; \backslash alpha^2\}\; =\; m\; -\; q\; +\; \backslash sum\_\{i=1\}^q\; \backslash frac\{\backslash alpha^2\}\{\backslash sigma\; \_i^2\; +\; \backslash alpha^2\}.$$

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Relation to probabilistic formulation The probabilistic formulation of an inverse problem introduces (when all uncertainties are Gaussian) a covariance matrix $C\_M$ representing the a priori uncertainties on the model parameters, and a covariance matrix $C\_D$ representing the uncertainties on the observed parameters.

(2025). 9780898717921, Society for Industrial and Applied Mathematics (SIAM). . ISBN 9780898717921

In the special case when these two matrices are diagonal and isotropic, $C\_M\; =\; \backslash sigma\_M^2\; I$ and $C\_D\; =\; \backslash sigma\_D^2\; I$, and, in this case, the equations of inverse theory reduce to the equations above, with $\backslash alpha\; =\; \{\backslash sigma\_D\}/\{\backslash sigma\_M\}$.

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Bayesian interpretation Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix $\backslash Gamma$ seems rather arbitrary, the process can be justified from a Bayesian point of view.

(1983). 9780471090779, John Wiley & Sons. ISBN 9780471090779

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 $x$ is sometimes taken to be a multivariate normal distribution. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation $\backslash sigma\; \_x$. The data are also subject to errors, and the errors in $b$ are also assumed to be independent with zero mean and standard deviation $\backslash sigma\; \_b$. Under these assumptions the Tikhonov-regularized solution is the most probable solution given the data and the a priori distribution of $x$, according to Bayes' theorem.

Vogel, Curtis R. (2025). 9780898715507, Society for Industrial and Applied Mathematics. ISBN 9780898715507

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.

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See also

• LASSO estimator is another regularization method in statistics.
• Elastic net regularization
• Matrix regularization
• L-curve

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Notes

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Further reading

• Marvin Gruber (1998). 9780824701567, CRC Press. . ISBN 9780824701567
• Rainer Kress (1998). 9780387984087, Springer. ISBN 9780387984087
• (2025). 9780521880688, Cambridge University Press. ISBN 9780521880688
• (2025). 9781118644614, John Wiley & Sons. . ISBN 9781118644614
• Matt Taddy (2025). 9781260452778, McGraw-Hill. ISBN 9781260452778

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