Tikhonov regularization

Tikhonov regularization, named for Andrey Tikhonov, is a method of regularization of ill-posed problems. Also known as ridge regression,[a] 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).

In the simplest case, the problem of a near-singular moment matrix $(\mathbf {X} ^{\mathsf {T}}\mathbf {X} )$ is alleviated by adding positive elements to the diagonals. The approach can be conceptualized by posing a constraint $\sum \beta _{i}^{2}=c$ to the least squares problem, such that

$\min _{\beta }\,(\mathbf {y} -\mathbf {X} \beta )^{\mathsf {T}}(\mathbf {y} -\mathbf {X} \beta )+\lambda (\beta ^{\mathsf {T}}\beta -c)$ where $\lambda$ is the Lagrange multiplier of the constraint. The minimizer of the problem is the simple ridge estimator

${\hat {\beta }}_{R}=(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +\lambda \mathbf {I} )^{-1}\mathbf {X} ^{\mathsf {T}}\mathbf {y}$ where $\mathbf {I}$ is the identity matrix and the ridge parameter $\lambda$ serves as the positive constant shifting the diagonals, thereby decreasing the condition number of the moment matrix. A more general approach to Tikhonov regularization is discussed below.

History

Tikhonov regularization has been invented independently in many different contexts. It became widely known from its application to integral equations from the work of Andrey Tikhonov 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 Wiener–Kolmogorov (Kriging) filter. Following Hoerl, it is known in the statistical literature as ridge regression.

Tikhonov regularization

Suppose that for a known matrix $A$ and vector $\mathbf {b}$ , we wish to find a vector $\mathbf {x}$ such that

$A\mathbf {x} =\mathbf {b} .$ The standard approach is ordinary least squares linear regression. However, if no $\mathbf {x}$ satisfies the equation or more than one $\mathbf {x}$ 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 $\mathbf {x}$ to $\mathbf {b}$ . 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 $\mathbf {x}$ that is in the null-space of $A$ , rather than allowing for a model to be used as a prior for $\mathbf {x}$ . Ordinary least squares seeks to minimize the sum of squared residuals, which can be compactly written as

$\|A\mathbf {x} -\mathbf {b} \|_{2}^{2},$ where $\|\cdot \|_{2}$ 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:

$\|A\mathbf {x} -\mathbf {b} \|_{2}^{2}+\|\Gamma \mathbf {x} \|_{2}^{2}$ for some suitably chosen Tikhonov matrix $\Gamma$ . In many cases, this matrix is chosen as a multiple of the identity matrix ($\Gamma =\alpha I$ ), giving preference to solutions with smaller norms; this is known as L2 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 ${\hat {x}}$ , is given by

${\hat {x}}=(A^{\top }A+\Gamma ^{\top }\Gamma )^{-1}A^{\top }\mathbf {b} .$ The effect of regularization may be varied by the scale of matrix $\Gamma$ . For $\Gamma =0$ this reduces to the unregularized least-squares solution, provided that (ATA)−1 exists.

L2 regularization is used in many contexts aside from linear regression, such as classification with logistic regression or support vector machines, and matrix factorization.

Generalized Tikhonov regularization

For general multivariate normal distributions for $x$ and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an $x$ to minimize

$\|Ax-b\|_{P}^{2}+\|x-x_{0}\|_{Q}^{2},$ where we have used $\|x\|_{Q}^{2}$ to stand for the weighted norm squared $x^{\top }Qx$ (compare with the Mahalanobis distance). In the Bayesian interpretation $P$ is the inverse covariance matrix of $b$ , $x_{0}$ is the expected value of $x$ , and $Q$ is the inverse covariance matrix of $x$ . The Tikhonov matrix is then given as a factorization of the matrix $Q=\Gamma ^{\top }\Gamma$ (e.g. the Cholesky factorization) and is considered a whitening filter.

This generalized problem has an optimal solution $x^{*}$ which can be written explicitly using the formula

$x^{*}=(A^{\top }PA+Q)^{-1}(A^{\top }Pb+Qx_{0}),$ or equivalently

$x^{*}=x_{0}+(A^{\top }PA+Q)^{-1}(A^{\top }P(b-Ax_{0})).$ Lavrentyev regularization

In some situations, one can avoid using the transpose $A^{\top }$ , as proposed by Mikhail Lavrentyev. For example, if $A$ is symmetric positive definite, i.e. $A=A^{\top }>0$ , so is its inverse $A^{-1}$ , which can thus be used to set up the weighted norm squared $\|x\|_{P}^{2}=x^{\top }A^{-1}x$ in the generalized Tikhonov regularization, leading to minimizing

$\|Ax-b\|_{A^{-1}}^{2}+\|x-x_{0}\|_{Q}^{2}$ or, equivalently up to a constant term,

$x^{\top }(A+Q)x-2x^{\top }(b+Qx)$ .

This minimization problem has an optimal solution $x^{*}$ which can be written explicitly using the formula

$x^{*}=(A+Q)^{-1}(b+Qx_{0})$ ,

which is nothing but the solution of the generalized Tikhonov problem where $A=A^{\top }=P^{-1}.$ 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 $A^{\top }A+\Gamma ^{\top }\Gamma .$ 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 Hilbert spaces, and $x$ and $b$ as elements in the domain and range of $A$ . The operator $A^{*}A+\Gamma ^{\top }\Gamma$ is then a self-adjoint bounded invertible operator.

Relation to singular-value decomposition and Wiener filter

With $\Gamma =\alpha I$ , this least-squares solution can be analyzed in a special way using the singular-value decomposition. Given the singular value decomposition

$A=U\Sigma V^{\top }$ with singular values $\sigma _{i}$ , the Tikhonov regularized solution can be expressed as

${\hat {x}}=VDU^{\top }b,$ where $D$ has diagonal values

$D_{ii}={\frac {\sigma _{i}}{\sigma _{i}^{2}+\alpha ^{2}}}$ 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.

Finally, it is related to the Wiener filter:

${\hat {x}}=\sum _{i=1}^{q}f_{i}{\frac {u_{i}^{\top }b}{\sigma _{i}}}v_{i},$ where the Wiener weights are $f_{i}={\frac {\sigma _{i}^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}$ and $q$ is the rank of $A$ .

Determination of the Tikhonov factor

The optimal regularization parameter $\alpha$ 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, 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

$G={\frac {\operatorname {RSS} }{\tau ^{2}}}={\frac {\|X{\hat {\beta }}-y\|^{2}}{[\operatorname {Tr} (I-X(X^{T}X+\alpha ^{2}I)^{-1}X^{T})]^{2}}},$ where $\operatorname {RSS}$ is the residual sum of squares, and $\tau$ is the effective number of degrees of freedom.

Using the previous SVD decomposition, we can simplify the above expression:

$\operatorname {RSS} =\left\|y-\sum _{i=1}^{q}(u_{i}'b)u_{i}\right\|^{2}+\left\|\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}(u_{i}'b)u_{i}\right\|^{2},$ $\operatorname {RSS} =\operatorname {RSS} _{0}+\left\|\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}(u_{i}'b)u_{i}\right\|^{2},$ and

$\tau =m-\sum _{i=1}^{q}{\frac {\sigma _{i}^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}=m-q+\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}.$ 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. In the special case when these two matrices are diagonal and isotropic, $C_{M}=\sigma _{M}^{2}I$ and $C_{D}=\sigma _{D}^{2}I$ , and, in this case, the equations of inverse theory reduce to the equations above, with $\alpha ={\sigma _{D}}/{\sigma _{M}}$ .

Bayesian interpretation

Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix $\Gamma$ seems rather arbitrary, the process can be justified from a Bayesian point of view. 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 $\sigma _{x}$ . The data are also subject to errors, and the errors in $b$ are also assumed to be independent with zero mean and standard deviation $\sigma _{b}$ . 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.

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 estimator.