Tikhonov regularization
From Wikipedia, the free encyclopedia
Tikhonov regularization is the most commonly used method of regularization of ill-posed problems named for Andrey Tychonoff. In statistics, the method is also known as ridge regression. It is related to the Levenberg-Marquardt algorithm for non-linear least-squares problems.
The standard approach to solve an underdetermined system of linear equations given as
is known as linear least squares and seeks to minimize the residual
where is the Euclidean norm. However, the matrix A may be ill-conditioned or singular yielding a non-unique solution. 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, . In many cases, this matrix is chosen as the identity matrix = 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 . For = αI, when α = 0 this reduces to the unregularized least squares solution provided that (ATA)−1 exists.
Contents
∙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
Bayesian interpretation
Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix 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 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 σx. Our data is also subject to errors, and we take the errors in b to be also independent with zero mean and standard deviation σb. 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 matrix is then = αI for Tikhonov factor α = σb / σx.
If the assumption of normality is replaced by assumptions of homoskedasticity and uncorrelatedness of errors, and still assume zero mean, then the Gauss-Markov theorem entails that the solution is 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 xTPx (cf. the Mahalanobis distance). In the Bayesian interpretation P is the inverse covariance matrix of b, x0 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 = T(e.g. the cholesky factorization), and is considered a whitening filter.
This generalized problem can be solved explicitly using the formula
[edit] Regularization in Hilbert space
Typically discrete linear ill-conditioned 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 is then a self-adjoint bounded invertible operator.
Relation to singular value decomposition and Wiener filter
With = 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 σi, 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 正则化英文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 i
s 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, 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:

版权声明：本站内容均来自互联网，仅供演示用，请勿用于商业和其他非法用途。如果侵犯了您的权益请与我们联系QQ:729038198，我们将在24小时内删除。