Convergence of generalized cross-validation with applications to ill-posed integral equations

2025/06/17 by Tim Jahn, Jahn, Tim, Mikhail Kirilin +1
Engineering · Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Microwave Imaging and Scattering Analysis #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2506.14558

openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/08/01

Abstract

In this article, we rigorously establish the consistency of generalized cross-validation as a parameter-choice rule for solving inverse problems. We prove that the index chosen by leave-one-out GCV achieves a non-asymptotic, order-optimal error bound with high probability for polynomially ill-posed compact operators. Hereby it is remarkable that the unknown true solution need not satisfy a self-similarity condition, which is generally needed for other heuristic parameter choice rules. We quantify the rate and demonstrate convergence numerically on integral equation test cases, including image deblurring and CT reconstruction.

Citations

• Asymptotically free sketched ridge ensembles: Risks, cross-validation, and tuning
• A Probabilistic Oracle Inequality and Quantification of Uncertainty of a modified Discrepancy Principle for Statistical Inverse Problems
• Optimal Convergence of the Discrepancy Principle for polynomially and exponentially ill-posed Operators under White Noise
• IR Tools: A MATLAB Package of Iterative Regularization Methods and Large-Scale Test Problems
• Empirical Risk Minimization as Parameter Choice Rule for General Linear\n Regularization Methods
• Risk Estimators for Choosing Regularization Parameters in Ill-Posed\n Problems - Properties and Limitations
• Optimal adaptation for early stopping in statistical inverse problems
• Bayes procedures for adaptive inference in inverse problems for the white noise model
• A survey of cross-validation procedures for model selection
• Smoothing noisy data with spline functions

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• ℓ¹ penalty for ill-posed inverse problems
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