Tikhonov regularization is the most commonly used method of regularization of ill-posed problems. In its simplest form, an ill-conditioned system of linear equations The mathematical term regularization has two main meanings, both associated with making a function more `regular or smooth. ... In mathematics, an ill-posed problem is one that is not well-posed, in that it violates one or more of the following conditions: A solution exists. ... In numerical analysis, the condition number associated with a numerical problem is a measure of that quantitys amenability to digital computation, that is, how well-posed the problem is. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ...

Ax = b,

where A is an m×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 For the square matrix section, see square matrix. ...

||Ax − b||² + α² ||x||²

for some suitably chosen Tikhonov factor α > 0. Here ||.|| is the Euclidean norm. In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

This improves the conditioning of the problem, thus enabling a numerical solution. An explicit solution is given by

(A^TA + α² I)⁻¹ A^Tb

where I is the n×n identity matrix. For α = 0 this reduces to the least squares solution of an overdetermined problem (m > n). In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... Linear least squares is a mathematical optimization technique to find an approximate solution for a system of linear equations that has no exact solution. ...

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 α = σ_b/σ_x. Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of a statement. ... In mathematics, an ill-posed problem is one that is not well-posed, in that it violates one or more of the following conditions: A solution exists. ... A priori is a Latin phrase meaning from the former (a, from, prior, prius, that which is before, precedes) or less literally before experience. It is used popularly of a judgment based on general considerations in the absence of particular evidence. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution (in honor of Carl Friedrich Gauss, who was not the first to write about the normal distribution) is a specific probability distribution. ... In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. ... In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. ... Bayes theorem is a result in probability theory. ...

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. The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In statistics, a sequence or a vector of random variables is homoscedastic if all random variables in the sequence or vector have the same finite variance. ... In statistics, the concepts of error and residual are easily confused with each other. ... This article is not about Gauss-Markov processes. ...

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 ||x||_P to stand for the weighted norm x^TPx. 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. In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical...

This can be solved explicitly using the formula

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 + α² I is then a self-adjoint bounded invertible operator for α > 0. In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, the Hermitian adjoint of an linear operator is a matching operator (very similar to the inverse operator in concept) defined over a linear space with inner product. ...

Relation to singular value decomposition

Given the singular value decomposition In linear algebra singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

A = UΣ V^T

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

V D U^T b

where D is an m-by-n matrix equal to

σ_i/(σ_i² + α²)

on the diagonal and zero elsewhere. This demonstrates the effect of the Tikhonov parameter on the condition number of the regularized problem. In numerical analysis, the condition number associated with a numerical problem is a measure of that quantitys amenability to digital computation, that is, how well-posed the problem is. ...

For the generalized case a similar representation can be derived using a generalized singular value decomposition.

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. Andrey Nikolayevich Tychonoff (Андрей Николаевич Тихонов: October 30, 1906–1993) was a Russian mathematician. ... Norbert Wiener Norbert Wiener (November 26, 1894 - March 18, 1964) was an American mathematician, known as the founder of cybernetics. ... Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ...

References

• Tikhonov AN, 1943, On the stability of inverse problems, Dokl. Akad. Nauk SSSR, 39, No. 5, 195-198
• Hoerl AE, 1962, Application of ridge analysis to regression problems. Chemical Engineering Progress, 58, 54-59.
• Foster, Manus, 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
• 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 0470991240.
• Tarantola A, 1987, Inverse Problem Theory, Elsevier, ISBN 0444427651.