Generalized cross validation for ℓp-ℓq minimization

Abstract

Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p-norm, and a regularization term, that is defined in terms of a q-norm. We allow 0 < p,q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed.

A comparison of parameter choice rules for $$\ell ^p$$-$$\ell ^q$$ minimization

“…We refer to the solution subspace V k+1 = range(V k+1 ) as a generalized Krylov subspace. Note that the computation of r (k+1) requires only one matrix-vector product with A T and L T , since we can exploit the QR factorizations (11) and the relation ( 8) to avoid computing any other matrix-vector products with the matrices A and L. Moreover, we store and update the "skinny" matrices AV k and LV k at each iteration to reduce the computational cost. The initial space V 1 is usually chosen to contain a few selected vectors and to be of small dimension.…”

“…This subsection summarizes the approach presented in [11]. Similarly as above, we consider a non-stationary method for determining μ.…”

“…When the GSVD of the matrix pair {A, L} is available, G(μ) can be evaluated inexpensively. If the matrices A and L are large, then they can be reduced to smaller matrices by a Krylov-type method and the GCV method can be applied to the reduced problem so obtained; see [11] for more details. This is how we apply the GCV method in the methods of the present paper.…”

“…We now consider the case in which the noise is not Gaussian. Since, in the continuous setting, the value of Ax μ − b δ 2 is infinite when b δ is corrupted by impulse noise, a smoothed version of GCV was proposed in [11]. Let b δ smooth denote a smoothed version of b δ obtained by convolving b δ with a Gaussian kernel; see [11] for details.…”

An $\ell_p$ Variable Projection Method for Large-Scale Separable Nonlinear Inverse Problems

“…However, computing the SVD might not be computationally attractive or might even prohibitive for large scale problems, and therefore, they also computed the GCV function by the aid of a Lanczos type method. Other references where efficient ways to compute the GCV can be found in [6,19,20]. For problems of the form like (2.1) with p = 2, GCV chooses the value λ that minimizes the function…”