In inverse problems, it is widely recognized that the incorporation of a sparsity prior yields a regularization effect on the solution. This approach is grounded on the a priori assumption that the unknown can be appropriately represented in a basis with a limited number of significant components, while most coefficients are close to zero. This occurrence is frequently observed in real-world scenarios, such as with piecewise smooth signals. In this study, we propose a probabilistic sparsity prior formulated as a mixture of degenerate Gaussians, capable of modeling sparsity with respect to a generic basis. Under this premise, we design a neural network that can be interpreted as the Bayes estimator for linear inverse problems. Additionally, we put forth both a supervised and an unsupervised training strategy to estimate the parameters of this network. To evaluate the effectiveness of our approach, we conduct a numerical comparison with commonly employed sparsity-promoting regularization techniques, namely LASSO, group LASSO, iterative hard thresholding, and sparse coding/dictionary learning. Notably, our reconstructions consistently exhibit lower mean square error values across all $1$D datasets utilized for the comparisons, even in cases where the datasets significantly deviate from a Gaussian mixture model.

翻译：在逆问题中，广泛认可的是，引入稀疏性先验会对解产生正则化效果。该方法基于一个先验假设，即未知量可以在某个基上用有限数量的显著分量适当表示，而大多数系数接近零。这一现象在现实场景中经常出现，例如分段平滑信号。在本研究中，我们提出了一种概率稀疏性先验，其形式为退化高斯混合模型，能够针对通用基对稀疏性进行建模。基于这一前提，我们设计了一个神经网络，该网络可被解释为线性逆问题的贝叶斯估计器。此外，我们提出了监督和无监督两种训练策略来估计该网络的参数。为了评估我们方法的有效性，我们与常用的稀疏促进正则化技术（即LASSO、组LASSO、迭代硬阈值和稀疏编码/字典学习）进行了数值比较。值得注意的是，在所有用于比较的1D数据集上，我们的重建结果始终表现出较低的均方误差值，即使数据集与高斯混合模型存在显著偏差时也是如此。