We propose a class of greedy algorithms for weighted sparse recovery by considering new loss function-based generalizations of Orthogonal Matching Pursuit (OMP). Given a (regularized) loss function, the proposed algorithms alternate the iterative construction of the signal support via greedy index selection and a signal update based on solving a local data-fitting problem restricted to the current support. We show that greedy selection rules associated with popular weighted sparsity-promoting loss functions admit explicitly computable and simple formulas. Specifically, we consider $ \ell^0 $- and $ \ell^1 $-based versions of the weighted LASSO (Least Absolute Shrinkage and Selection Operator), the Square-Root LASSO (SR-LASSO) and the Least Absolute Deviations LASSO (LAD-LASSO). Through numerical experiments on Gaussian compressive sensing and high-dimensional function approximation, we demonstrate the effectiveness of the proposed algorithms and empirically show that they inherit desirable characteristics from the corresponding loss functions, such as SR-LASSO's noise-blind optimal parameter tuning and LAD-LASSO's fault tolerance. In doing so, our study sheds new light on the connection between greedy sparse recovery and convex relaxation.

翻译：我们提出了一类针对加权稀疏恢复的贪婪算法，通过考虑基于损失函数的正交匹配追踪（OMP）新推广形式。给定一个（正则化）损失函数，所提算法交替进行：通过贪婪索引选取迭代构建信号支撑集，以及基于求解限制在当前支撑集上的局部数据拟合问题进行信号更新。我们证明，与流行的加权稀疏性促进损失函数相关的贪婪选择规则存在可显式计算且形式简洁的公式。具体而言，我们考虑了基于 $\ell^0$ 和 $\ell^1$ 范数的加权LASSO（最小绝对收缩与选择算子）、平方根LASSO（SR-LASSO）以及最小绝对偏差LASSO（LAD-LASSO）版本。通过在高斯压缩感知和高维函数逼近问题上的数值实验，我们验证了所提算法的有效性，并经验性地表明它们继承了相应损失函数的优良特性，例如SR-LASSO的噪声盲自适应最优参数调节能力和LAD-LASSO的容错性。本研究为贪婪稀疏恢复与凸松弛之间的关联提供了新见解。