Learning a Gaussian Mixture for Sparsity Regularization in Inverse
  Problems
Giovanni S. Alberti Luca Ratti Matteo Santacesaria Silvia Sciutto
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.