基于分层稀疏促进先验的保边缘贝叶斯神经网络
Edge-preserving Bayesian neural networks via hierarchical sparsity-promoting priors
AI总结：
该研究针对含噪不完整数据的反问题，提出GSBL-BNNs模型，通过分层稀疏先验与BCD算法实现保边缘重建及UQ，性能优于高斯、柯西-BNNs。
摘要：
从含噪、间接且不完整的数据中恢复带有跳跃间断的分段光滑函数，是计算反问题中的核心任务。带有重尾权重先验的贝叶斯神经网络（BNNs）已成为这类问题的有效保边缘先验，但这些模型的后验推断仍存在计算挑战，现有方法仅满足于通过直接优化重尾对数密度得到的点估计。我们提出GSBL-BNNs，其利用广义稀疏贝叶斯学习（GSBL）中的稀疏促进先验对神经网络参数进行分层建模，该分层公式可恢复拉普拉斯、学生t（含柯西）等多种流行的边缘先验。保边缘BNNs中一种新颖的关键建模要素（源自该分层公式），通过共享超参数将每个神经元的偏置与其权重耦合，使得诱导的跳跃间断可在定义域内任意位置出现，同时保留高效推断所需的条件高斯结构。此外，我们开发了用于最大后验（MAP）估计的块坐标下降（BCD）算法，该BCD方法在带迭代重加权ℓ²正则化的标准网络训练与伽马、逆伽马超先验的精确闭式超参数更新之间交替进行。基于该BCD算法，我们利用深度集成执行近似贝叶斯推断与不确定性量化（UQ）。计算实验包括信号去噪、去模糊及二维双曲守恒律的反演，结果表明GSBL-BNNs可生成准确的保边缘重建结果并内置UQ，与高斯-BNNs和柯西-BNNs相比，其预测均值更准确且避免了过度自信的不确定性估计。
英文摘要：
Recovering piecewise-smooth functions with jump discontinuities from noisy, indirect, and incomplete data is a central task in computational inverse problems. Bayesian neural networks (BNNs) with heavy-tailed weight priors have emerged as effective edge-preserving priors for such problems. However, posterior inference for these models remains computationally challenging, and existing approaches settle for point estimates obtained by directly optimizing heavy-tailed log-densities. We propose GSBL-BNNs, which model the neural network parameters hierarchically using sparsity-promoting priors from generalized sparse Bayesian learning (GSBL). The hierarchical formulation recovers several popular marginal priors, including Laplace and Student's $t$ (including Cauchy). A key modeling ingredient, novel among edge-preserving BNNs and native to the hierarchical formulation, couples each neuron's bias to its weights through a shared hyperparameter, so that the induced jump discontinuities are free to occur anywhere in the domain while the conditionally Gaussian structure underlying efficient inference is preserved. Furthermore, we develop a block-coordinate descent (BCD) algorithm for maximum a posteriori (MAP) estimation. This BCD approach alternates between standard network training with iteratively reweighted $\ell^2$-regularization and exact, closed-form hyperparameter updates for gamma and inverse gamma hyperpriors. Building on the BCD algorithm, we perform approximate Bayesian inference and uncertainty quantification (UQ) using deep ensembles. Computational experiments---including signal denoising, deblurring, and inverting a two-dimensional hyperbolic conservation law--- show that GSBL-BNNs deliver accurate, edge-preserving reconstructions with built-in UQ. Compared with Gaussian- and Cauchy-BNNs, they yield more accurate predictive means and avoid overconfident uncertainty estimates.
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