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。

逆问题中的稀疏正则化中学习高斯混合模型

Learning a Gaussian Mixture for Sparsity Regularization in Inverse  Problems

We study the computational complexity of Markov chain Monte Carlo (MCMC)
methods for high-dimensional Bayesian linear regression under sparsity
constraints. We first show that a Bayesian approach can achieve
variable-selection consistency under relatively mild conditions on the design
matrix. We then demonstrate that the statistical criterion of posterior
concentration need not imply the computational desideratum of rapid mixing of
the MCMC algorithm. By introducing a truncated sparsity prior for variable
selection, we provide a set of conditions that guarantee both
variable-selection consistency and rapid mixing of a particular
Metropolis-Hastings algorithm. The mixing time is linear in the number of
covariates up to a logarithmic factor. Our proof controls the spectral gap of
the Markov chain by constructing a canonical path ensemble that is inspired by
the steps taken by greedy algorithms for variable selection.

本文研究高维贝叶斯线性回归的计算复杂度，介绍了一种截尾稀疏先验变量选择方法，通过 Metropolis-Hastings 算法，保证了变量选择的一致性和快速混合。

高维贝叶斯变量选择的计算复杂度

On the Computational Complexity of High-Dimensional Bayesian Variable  Selection

Compressed sensing (CS) is an emerging field that has attracted considerable
research interest over the past few years. Previous review articles in CS limit
their scope to standard discrete-to-discrete measurement architectures using
matrices of randomized nature and signal models based on standard sparsity. In
recent years, CS has worked its way into several new application areas. This,
in turn, necessitates a fresh look on many of the basics of CS. The random
matrix measurement operator must be replaced by more structured sensing
architectures that correspond to the characteristics of feasible acquisition
hardware. The standard sparsity prior has to be extended to include a much
richer class of signals and to encode broader data models, including
continuous-time signals. In our overview, the theme is exploiting signal and
measurement structure in compressive sensing. The prime focus is bridging
theory and practice; that is, to pinpoint the potential of structured CS
strategies to emerge from the math to the hardware. Our summary highlights new
directions as well as relations to more traditional CS, with the hope of
serving both as a review to practitioners wanting to join this emerging field,
and as a reference for researchers that attempts to put some of the existing
ideas in perspective of practical applications.

本文论述了在压缩感知中利用信号和测量结构进行体现的主题，并将理论与实践相结合，介绍了新的方向及与传统压缩感知的关系。