Adaptive regularization parameter selection for high-dimensional inverse problems: A Bayesian approach with Tucker low-rank constraints
arXiv:2603.16066v1 Announce Type: new Abstract: This paper introduces a novel variational Bayesian method that integrates Tucker decomposition for efficient high-dimensional inverse problem solving. The method reduces computational complexity by transforming variational inference from a high-dimensional space to a lower-dimensional core tensor space via Tucker decomposition. A key innovation is the introduction of per-mode precision parameters, enabling adaptive regularization for anisotropic structures. For instance, in directional image deblurring, learned parameters align with physical anisotropy, applying stronger regularization to critical directions (e.g., row vs. column axes). The method further estimates noise levels from data, eliminating reliance on prior knowledge of noise parameters (unlike conventional benchmarks such as the discrepancy principle (DP)). Experimental evaluations across 2D deblurring, 3D heat conduction, and Fredholm integral equations demonstrate consisten
arXiv:2603.16066v1 Announce Type: new Abstract: This paper introduces a novel variational Bayesian method that integrates Tucker decomposition for efficient high-dimensional inverse problem solving. The method reduces computational complexity by transforming variational inference from a high-dimensional space to a lower-dimensional core tensor space via Tucker decomposition. A key innovation is the introduction of per-mode precision parameters, enabling adaptive regularization for anisotropic structures. For instance, in directional image deblurring, learned parameters align with physical anisotropy, applying stronger regularization to critical directions (e.g., row vs. column axes). The method further estimates noise levels from data, eliminating reliance on prior knowledge of noise parameters (unlike conventional benchmarks such as the discrepancy principle (DP)). Experimental evaluations across 2D deblurring, 3D heat conduction, and Fredholm integral equations demonstrate consistent improvements in quantitative metrics (PSNR, SSIM) and qualitative visualizations (error maps, precision parameter trends) compared to L-curve criterion, generalized cross-validation (GCV), unbiased predictive risk estimator (UPRE), and DP. The approach scales to problems with 110,000 variables and outperforms existing methods by 0.73-2.09 dB in deblurring tasks and 6.75 dB in 3D heat conduction. Limitations include sensitivity to rank selection in Tucker decomposition and the need for theoretical analysis. Future work will explore automated rank selection and theoretical guarantees. This method bridges Bayesian theory and scalable computation, offering practical solutions for large-scale inverse problems in imaging, remote sensing, and scientific computing.
Executive Summary
This article presents a novel variational Bayesian method for high-dimensional inverse problem solving, leveraging Tucker decomposition to reduce computational complexity and enable adaptive regularization. The method learns anisotropic structures and noise levels from data, eliminating prior knowledge of noise parameters. Experimental evaluations demonstrate consistent improvements in quantitative metrics and qualitative visualizations compared to existing methods. While the approach scales to large-scale problems and offers practical solutions for imaging, remote sensing, and scientific computing, it is sensitive to rank selection in Tucker decomposition and requires theoretical analysis. Future work will explore automated rank selection and theoretical guarantees.
Key Points
- ▸ Adaptive regularization parameter selection using per-mode precision parameters
- ▸ Tucker low-rank constraints for reducing computational complexity
- ▸ Noise level estimation from data, eliminating prior knowledge of noise parameters
Merits
Strength in addressing anisotropic structures
The method leverages per-mode precision parameters to adaptively regularize for anisotropic structures, leading to improved performance in directional image deblurring.
Demerits
Sensitivity to rank selection in Tucker decomposition
The method is sensitive to the choice of rank in Tucker decomposition, which can impact its performance and scalability.
Expert Commentary
The article presents a significant contribution to the field of Bayesian inference in high-dimensional spaces, leveraging the strengths of Tucker decomposition to reduce computational complexity and enable adaptive regularization. The experimental evaluations demonstrate the method's efficacy in a range of applications, from directional image deblurring to 3D heat conduction. However, the method's sensitivity to rank selection in Tucker decomposition and the need for theoretical analysis are notable limitations. Future work should focus on developing automated rank selection methods and theoretical guarantees to further establish the method's robustness and reliability.
Recommendations
- ✓ Future research should explore the application of this method to other high-dimensional inverse problems in imaging, remote sensing, and scientific computing.
- ✓ Theoretical analysis should be conducted to provide a deeper understanding of the method's performance and scalability, including a rigorous examination of its sensitivity to rank selection in Tucker decomposition.