 Tikhonov regularization

Tikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of illposed problems. In statistics, the method is known as ridge regression, and, with multiple independent discoveries, it is also variously known as the TikhonovMiller method, the PhillipsTwomey method, the constrained linear inversion method, and the method of linear regularization. It is related to the LevenbergMarquardt algorithm for nonlinear leastsquares problems.
When the following problem is not well posed (either because of nonexistence or nonuniqueness of x)
then the standard approach is known as linear least squares and seeks to minimize the residual
where is the Euclidean norm. This may be due to the system being overdetermined or underdetermined (A may be illconditioned or singular). In the latter case this is no better than the original problem. In order to give preference to a particular solution with desirable properties, the regularization term is included in this minimization:
for some suitably chosen Tikhonov matrix, Γ. In many cases, this matrix is chosen as the identity matrix Γ = I, giving preference to solutions with smaller norms. In other cases, highpass 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 numerical solution. An explicit solution, denoted by , is given by:
The effect of regularization may be varied via the scale of matrix Γ. For Γ = αI, when α = 0 this reduces to the unregularized least squares solution provided that (A^{T}A)^{−1} exists.
Contents
Bayesian interpretation
Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix Γ seems rather arbitrary, the process can be justified from a Bayesian point of view. Note that for an illposed 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 are 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 Tikhonovregularized solution is the most probable solution given the data and the a priori distribution of x, according to Bayes' theorem. The Tikhonov matrix is then Γ = αI for Tikhonov factor α = σ_{b} / σ_{x}.
If the assumption of normality is replaced by assumptions of homoskedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the GaussMarkov theorem entails that the solution is the minimal unbiased estimate.
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
where we have used to stand for the weighted norm x^{T}Qx (cf. 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 = Γ^{T}Γ (e.g. the Cholesky factorization), and is considered a whitening filter.
This generalized problem can be solved explicitly using the formula
Regularization in Hilbert space
Typically discrete linear illconditioned 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 + Γ^{T}Γ is then a selfadjoint bounded invertible operator.
Relation to singular value decomposition and Wiener filter
With Γ = αI, this least squares solution can be analyzed in a special way via the singular value decomposition. Given the singular value decomposition of A
with singular values σ_{i}, the Tikhonov regularized solution can be expressed as
where D has diagonal values
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:
where the Wiener weights are and q is the rank of A.
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, crossvalidation, Lcurve method, restricted maximum likelihood and unbiased predictive risk estimator. Grace Wahba proved that the optimal parameter, in the sense of leaveoneout crossvalidation minimizes:
where 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:
and
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, 2004 [1]). In the special case when these two matrices are diagonal and isotropic, and , and, in this case, the equations of inverse theory reduce to the equations above, with α = σ_{D} / σ_{M}.
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 Tychonoff and David L. Phillips. Some authors use the term TikhonovPhillips 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 WienerKolmogorov filter. Following Hoerl, it is known in the statistical literature as ridge regression.
References
 Tychonoff, Andrey Nikolayevich (1943). "Об устойчивости обратных задач [On the stability of inverse problems]". Doklady Akademii Nauk SSSR 39 (5): 195–198.
 Tychonoff, A. N. (1963). "О решении некорректно поставленных задач и методе регуляризации [Solution of incorrectly formulated problems and the regularization method]". Doklady Akademii Nauk SSSR 151: 501–504.. Translated in Soviet Mathematics 4: 1035–1038.
 Tychonoff, A. N.; V. Y. Arsenin (1977). Solution of Illposed Problems. Washington: Winston & Sons. ISBN 0470991240.
 Hansen, P.C., 1998, Rankdeficient and Discrete illposed problems, SIAM
 Hoerl AE, 1962, Application of ridge analysis to regression problems, Chemical Engineering Progress, 58, 5459.
 Hoerl, A.E.; R.W. Kennard (1970). "Ridge regression: Biased estimation for nonorthogonal problems". Technometrics 42 (1). JSTOR 1271436.
 Foster M, 1961, An application of the WienerKolmogorov smoothing theory to matrix inversion, J. SIAM, 9, 387392
 Phillips DL, 1962, A technique for the numerical solution of certain integral equations of the first kind, J Assoc Comput Mach, 9, 8497
 Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 19.4. Linear Regularization Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 9780521880688. http://apps.nrbook.com/empanel/index.html#pg=1006.
 Tarantola A, 2004, Inverse Problem Theory (free PDF version), Society for Industrial and Applied Mathematics, ISBN 0898715725
 Wahba, G, 1990, Spline Models for Observational Data, Society for Industrial and Applied Mathematics
See Also
 LASSO estimator is another regularization method in statistics.
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