Tikhonov吉洪诺夫正则化

Tikhonov regularization

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Tikhonov regularization is the most commonly used method of of named for . In , the method is also known as ridge regression . It is related to the for problems.

The standard approach to solve an of given as

,b Ax =

is known as and seeks to minimize the

2

b

Ax -

where •is the . However, the matrix A may be or 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 Γ= I , giving preference to solutions with smaller norms. In other cases, operators ., a or a weighted ) 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 1

ˆ-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.

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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 . 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 we know that x is a random variable with a . For simplicity we take the mean to be zero and assume that each component is

independent with σx. Our data is also subject to errors, and we take the errors

in b to be also with zero mean and standard deviation σb. Under these assumptions the Tikhonov-regularized solution is the solution given the data

and the a priori distribution of x, according to . The Tikhonov matrix is then Γ=

αI for Tikhonov factor α = σb/ σx.

If the assumption of is replaced by assumptions of and uncorrelatedness of , and still assume zero mean, then the entails that the solution is minimal . 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