Tikhonov regularization

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Tikhonov regularization is the most commonly used method of regularization of ill-posed problems. In some fields, it is also known as ridge regression.

In its simplest form, an ill-conditioned system of linear equations

$A\mathbf{x}=\mathbf{b}$ ,

where $A$ is an $m \times n$ matrix above, $x$ is a column vector with $n$ entries and $b$ is a column vector with $m$ entries, is replaced by the problem of seeking an $x$ to minimize

$\|A\mathbf{x}-\mathbf{b}\|^2+\alpha^2\|\mathbf{x}\|^2$

for some suitably chosen Tikhonov factor $α > 0$ . Here $\left \| \cdot \right \|$ is the Euclidean norm. This improves the conditioning of the problem, thus enabling a numerical solution. An explicit solution, denoted by $\hat{x}$ , is given by:

$\hat{x} = (A^{T}A+\alpha^2I)^{-1}A^{T}\mathbf{b}$

where $I$ is the $n \times n$ identity matrix. For $α = 0$ this reduces to the least squares solution of an overdetermined problem ( $m > n$ ).

1 Bayesian interpretation
2 Generalized Tikhonov regularization
3 Regularization in Hilbert space
4 Relation to singular value decomposition and Wiener filter
5 Determination of the Tikhonov factor
6 Relation to probabilistic formulation
7 History
8 References

[edit] Bayesian interpretation

Although at first the choice of the solution to this regularized problem may look artificial, and indeed the parameter $α$ seems rather arbitrary, the process can be justified in a Bayesian point of view. Note that for an ill-posed problem one must necessarily introduce some additional assumptions in order to get a stable solution. Statistically we might assume that a priori we know that $x$ is a random variable with a multivariate normal distribution. For simplicity we take the mean to be zero and assume that each component is independent with standard deviation $σ x$ . Our data is also subject to errors, and we take the errors in $b$ to be also independent with zero mean and standard deviation $σ b$ . 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. The Tikhonov parameter is then $\alpha = \frac{\sigma _b}{\sigma _x}$ ...

If the assumption of normality is replaced by assumptions of homoscedasticity and uncorrelatedness of errors, and still assume zero mean, then the Gauss-Markov theorem entails that the solution is still optimal in a certain sense.

[edit] 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

$\|Ax-b\|_P^2 + \alpha^2\|x-x_0\|_Q^2\,$

where we have used $\left \| x \right \|_P$ to stand for the weighted norm $x T P x$ . In the Bayesian interpretation $P$ is the inverse covariance matrix of $b$ , $x 0$ is the expected value of $x$ , and $α Q$ is the inverse covariance matrix of $x$ .

This can be solved explicitly using the formula

$x_0 + (A^T PA + \alpha^2Q)^{-1} A^T P(b-Ax_0).\,$

[edit] Regularization in Hilbert space

Typically discrete linear ill-condition problems result as discretization of integral equations, and one can formulate 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 $A * A + α 2 I$ is then a self-adjoint bounded invertible operator for $α > 0$ .

[edit] Relation to singular value decomposition and Wiener filter

Given the singular value decomposition

A = U Σ V T

where $Σ$ is the diagonal matrix of singular values $σ i$ (augmented with zeros so as to be $m \times n$ ) and $U$ and $V$ respectively the matrices of left and right singular vectors then the Tikhonov regularized solution can be expressed as

$\hat{x} = V D U^T b$

where $D$ is an $m \times n$ matrix equal to

$\frac{\sigma _i}{\sigma _i ^2 + \alpha ^2}$

on the diagonal and 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:

$\hat{x} = \sum _{i=1} ^q f_i \frac{u_i ^T b}{\sigma _i} v_i$

where the Wiener weights are $f_i = \frac{\sigma _i ^2}{\sigma _i ^2 + \alpha ^2}$ and $q$ is the rank of $A$ .

[edit] Determination of the Tikhonov factor

The optimal regularization parameter $α$ is usually unknown and often in practical problems is determined by an ad hoc method. A possible approach relies on the Bayesian interpretation described above. Other approaches include the discrepancy principle, cross validation, L-curve method, and unbiased predictive risk estimator. Wahba proved that the optimal parameter, in the sense of leave-one-out cross-validation minimizes:

$G = \frac{\operatorname{RSS}}{\tau ^2} = \frac{\left \| X \hat{\beta} - y \right \| ^2}{\left[ \operatorname{Tr} \left(I - X (X^T X + \alpha I) ^{-1} X ^T \right) \right]^2}$

where $\operatorname{RSS}$ is the residual sum of squares and $τ$ is the effective number degree 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

$\tau = m - \sum _{i=1} ^q \frac{\sigma _i ^2}{\sigma _i ^2 + \alpha ^2} = m - q + \sum _{i=1} ^q \frac{\alpha ^2}{\sigma _i ^2 + \alpha ^2}$

[edit] 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 (see, for instance, Tarantola, 2005 [1]). 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 $α = σ D / σ M$ .

[edit] 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 AN Tikhonov and DL Phillips. Some authors use the term Tikhonov-Phillips regularization. The finite dimensional case was expounded by AE Hoerl, who took a statistical approach, and by M Foster, who interpreted this method as a Wiener-Kolmogorov filter. Following Hoerl, it is known in the statistical literature as ridge regression.

[edit] References

Tikhonov AN, 1943, On the stability of inverse problems, Dokl. Akad. Nauk SSSR, 39, No. 5, 195-198
Tikhonov AN, 1963, Solution of incorrectly formulated problems and the regularization method, Soviet Math Dokl 4, 1035-1038 English translation of Dokl Akad Nauk SSSR 151, 1963, 501-504
Tikhonov AN and Arsenin VA, 1977, Solution of Ill-posed Problems, Winston & Sons, Washington, ISBN 0-470-99124-0.
Hansen, P.C., Rank-deficient and Discrete ill-posed problems, SIAM
Hoerl AE, 1962, Application of ridge analysis to regression problems, Chemical Engineering Progress, 58, 54-59.
Foster M, 1961, An application of the Wiener-Kolmogorov smoothing theory to matrix inversion, J. SIAM, 9, 387-392
Phillips DL, 1962, A technique for the numerical solution of certain integral equations of the first kind, J Assoc Comput Mach, 9, 84-97
Tarantola A, 2005, Inverse Problem Theory (free PDF version), Society for Industrial and Applied Mathematics, ISBN 0-89871-572-5
Wahba, G, 1990, spline Models for Observational Data, SIAM