用贝叶斯方法自适应调节正则化，解决高维逆问题计算难问题。
Adaptive regularization parameter selection for high-dimensional inverse problems: A Bayesian approach with Tucker low-rank constraints
- 通过Tucker分解将高维问题转到低维核心张量空间，降低计算复杂度。
- 每模式精度参数自适应调节，对图像不同方向施加差异性正则化。
- 无需先验噪声信息，实测在去模糊和热传导任务中提升0.73-6.75dB。
本文提出一种新颖的变分贝叶斯方法，结合Tucker分解以高效求解高维逆问题。该方法通过Tucker分解将变分推断从高维空间转换至低维核心张量空间，显著降低计算复杂度。关键创新在于引入每模式精度参数，实现对各向异性结构的自适应正则化。例如，在方向性图像去模糊中，学习到的参数与物理各向异性一致，对关键方向（如行/列轴）施加强正则化。该方法可从数据中估计噪声水平，无需依赖噪声参数先验知识（不同于传统方法如残差准则）。在二维去模糊、三维热传导及弗雷德霍姆积分方程上的实验表明，相比L曲线法、广义交叉验证（GCV）、无偏预测风险估计（UPRE）和残差准则，本方法在定量指标（PSNR、SSIM）和定性结果（误差图、精度参数趋势）上均有持续提升。方法可扩展至11万变量的问题，在去模糊任务中性能领先0.73-2.09 dB，三维热传导任务中提升6.75 dB。局限性包括对Tucker分解秩选择敏感，且缺乏理论分析。未来工作将探索自动秩选择与理论保证。该方法连接贝叶斯理论与可扩展计算，为成像、遥感与科学计算中的大规模逆问题提供实用解决方案。
原文摘要 · Abstract (English)
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.
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