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.

本研究提出了一种概率稀疏性先验，用于建模相对于通用基的稀疏性，并设计了一个神经网络作为线性逆问题的贝叶斯估计器。通过与常用的稀疏性促进正则化技术进行比较，我们的重建方法在所有使用的一维数据集上均表现出更低的均方误差值。