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This paper presents a variational Bayesian method for solving high-dimensional inverse problems by integrating Tucker decomposition to reduce computational complexity and enable adaptive regularization. The approach introduces per-mode precision parameters, allowing for anisotropic regularization tailored to the specific problem structure, and estimates noise levels directly from the data. Experiments on 2D deblurring, 3D heat conduction, and Fredholm integral equations demonstrate improved performance compared to conventional regularization parameter selection methods, achieving gains of up to 6.75 dB in PSNR.
Forget grid search – this Bayesian method adaptively learns the optimal regularization parameters for each dimension in high-dimensional inverse problems, outperforming standard techniques by a significant margin.
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.