Tikhonov regularization

Tikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of ill-posed problems. In statistics, the method is known as ridge regression, and with multiple independent discoveries, it is also variously known as the Tikhonov–Miller method, the Phillips–Twomey method, the constrained linear inversion method, and the method of linear regularization. It is related to the Levenberg–Marquardt algorithm for non-linear least-squares problems.

When the following problem is not well posed (either because of non-existence or non-uniqueness of x)

A\mathbf{x}=\mathbf{b},

then the standard approach (known as ordinary least squares) leads to an overdetermined, or more often an underdetermined system of equations. Most real-world phenomena operate as low-pass filters in the forward direction where A maps \mathbf{x} to \mathbf{b}. Therefore in solving the inverse-problem, the inverse-mapping operates as a high-pass filter that has the undesirable tendency of amplifying noise (eigenvalues / singular values are largest in the reverse mapping where they were smallest in the forward mapping). In addition, ordinary least squares implicitly nullifies every element of the reconstructed version of \mathbf{x} that is in the null-space of A, rather than allowing for a model to be used as a prior for \mathbf{x}. Ordinary least squares seeks to minimize the sum of squared residuals, which can be compactly written as

\|A\mathbf{x}-\mathbf{b}\|^2

where \left \| \cdot \right \| is the Euclidean norm. In order to give preference to a particular solution with desirable properties, a regularization term can be included in this minimization:

\|A\mathbf{x}-\mathbf{b}\|^2+ \|\Gamma \mathbf{x}\|^2

for some suitably chosen Tikhonov matrix, \Gamma . In many cases, this matrix is chosen as a multiple of the identity matrix (\Gamma= \alpha I ), giving preference to solutions with smaller norms. In other cases, lowpass 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 direct numerical solution. An explicit solution, denoted by \hat{x}, is given by:

\hat{x} = (A^{T}A+ \Gamma^{T} \Gamma )^{-1}A^{T}\mathbf{b}

The effect of regularization may be varied via the scale of matrix \Gamma. For \Gamma = 0 this reduces to the unregularized least squares solution provided that (ATA)−1 exists.

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 Andrey Tikhonov and David L. Phillips. Some authors use the term Tikhonov–Phillips 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 Wiener–Kolmogorov filter. Following Hoerl, it is known in the statistical literature as ridge regression.

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

\|Ax-b\|_P^2 + \|x-x_0\|_Q^2\,

where we have used \left \| x \right \|_Q^2 to stand for the weighted norm x^T Q x (compare with 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 = \Gamma^T \Gamma (e.g. the Cholesky factorization), and is considered a whitening filter.

This generalized problem has an optimal solution x^* which can be solved explicitly using the formula

x^* = (A^T PA + Q)^{-1} (A^T Pb+Qx_0).\,

or equivalently

x^* = x_0 + (A^T PA + Q)^{-1} (A^T P(b-Ax_0)).\,

Regularization in Hilbert space

Typically discrete linear ill-conditioned 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 + \Gamma^T \Gamma is then a self-adjoint bounded invertible operator.

Relation to singular value decomposition and Wiener filter

With \Gamma = \alpha I , this least squares solution can be analyzed in a special way via the singular value decomposition. Given the singular value decomposition of A

A = U \Sigma V^T\,

with singular values \sigma _i, the Tikhonov regularized solution can be expressed as

\hat{x} = V D U^T b

where D has diagonal values

D_{ii} = \frac{\sigma _i}{\sigma _i ^2 + \alpha ^2}

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:

\hat{x} = \sum _{i=1} ^q f_i \frac{u_i ^T b}{\sigma _i} v_i

where the Wiener weights are f_i = \frac{\sigma _i ^2}{\sigma _i ^2 + \alpha ^2} and q is the rank of A.

Determination of the Tikhonov factor

The optimal regularization parameter \alpha is usually unknown and often in practical problems is determined by an ad hoc method. A possible approach relies on the Bayesian interpretation described below. 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:

G = \frac{\operatorname{RSS}}{\tau ^2} = \frac{\left \| X \hat{\beta} - y \right \| ^2}{\left[ \operatorname{Tr} \left(I - X (X^T X + \alpha ^2 I) ^{-1} X ^T \right) \right]^2}

where \operatorname{RSS} is the residual sum of squares and \tau is the effective number of degrees of freedom.

Using the previous SVD decomposition, we can simplify the above expression:

\operatorname{RSS} = \left \| y - \sum _{i=1} ^q (u_i ' b) u_i \right \| ^2 + \left \| \sum _{i=1} ^q \frac{\alpha ^ 2}{\sigma _i ^ 2 + \alpha ^ 2} (u_i ' b) u_i \right \| ^2
\operatorname{RSS} = \operatorname{RSS} _0 + \left \| \sum _{i=1} ^q \frac{\alpha ^ 2}{\sigma _i ^ 2 + \alpha ^ 2} (u_i ' b) u_i \right \| ^2

and

\tau = m - \sum _{i=1} ^q \frac{\sigma _i ^2}{\sigma _i ^2 + \alpha ^2} = m - q + \sum _{i=1} ^q \frac{\alpha ^2}{\sigma _i ^2 + \alpha ^2}

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, 2005 ). In the special case when these two matrices are diagonal and isotropic, C_M = \sigma_M^2 I and C_D = \sigma_D^2 I , and, in this case, the equations of inverse theory reduce to the equations above, with \alpha = {\sigma_D}/{\sigma_M} .

Bayesian interpretation

Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix \Gamma 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 unique solution. Statistically, the prior probability distribution of x is sometimes taken to be a multivariate normal distribution. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation \sigma _x. The data are also subject to errors, and the errors in b are also assumed to be independent with zero mean and standard deviation \sigma _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.[1]

If the assumption of normality is replaced by assumptions of homoskedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the Gauss–Markov theorem entails that the solution is the minimal unbiased estimator.

See also

References

  1. ↑ Vogel, Curtis R. (2002). Computational methods for inverse problems. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 0-89871-550-4.