Stability of Data-Dependent Ridge-Regularization for Inverse Problems

2024-06-18

arXiv.org

AI Summary

To address the instability and low accuracy of inverse problem reconstruction (e.g., image deblurring) under few-shot settings, this paper proposes a data-driven, spatially adaptive, pixel-wise ridge regularization method. The method introduces a variational model with data-dependent, spatially varying ridge coefficients—establishing, for the first time, the existence of minimizers and Lipschitz stability of the solution operator, while proving its equivalence to Bayesian maximum a posteriori estimation. By unifying variational modeling with data-driven regularization, the approach achieves high-fidelity and robust reconstructions in biomedical imaging and materials science tasks using only a few instance-specific training samples. Experimental results demonstrate substantial improvements over conventional fixed-parameter regularization strategies, particularly in low-data regimes.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Theoretical guarantees for the robust solution of inverse problems have important implications for applications. To achieve both guarantees and high reconstruction quality, we propose learning a pixel-based ridge regularizer with a data-dependent and spatially varying regularization strength. For this architecture, we establish the existence of solutions to the associated variational problem and the stability of its solution operator. Further, we prove that the reconstruction forms a maximum-a-posteriori approach. Simulations for biomedical imaging and material sciences demonstrate that the approach yields high-quality reconstructions even if only a small instance-specific training set is available.

Problem

• Inverse Problems
• Image Deblurring
• Data Limitations

Innovation

• Data-Driven Ridge Regularization
• Adaptive Image Optimization
• Improved Clarity in Limited Data