Tikhonov吉洪诺夫正则化

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

,b Ax = is known as linear least squares and seeks to minimize the residual 2b Ax - 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:

2

2x b Ax Γ+-

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:

()b A A A x T T T 1ˆ-ΓΓ+=

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

• 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 Γseems 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 Γ= 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.