This paper presents a greedy randomized average block sampling Gauss–Seidel (GRABGS) method for solving sparse linear least-squares problems. The GRABGS method utilizes a novel probability criterion to collect the control index set of coordinates, and minimizes the quadratic convex objective by performing multiple accurate line searches on average per iteration. The probability criterion aims to capture subvectors whose norms are relatively large. Additionly, the GRABGS method is categorized as a member of randomized block Gauss–Seidel methods, which can be employed for parallel implementations. The convergence analysis encompasses two types of extrapolation stepsizes: constant and adaptive. It is proved that the GRABGS method converges to the unique solution of the sparse linear least-squares problem when the matrix has full column rank. Numerical examples demonstrate the superiority of this method over the greedy randomized coordinate descent method and several existing state-of-the-art block Gauss–Seidel methods.

本文提出了一种求解稀疏线性最小二乘问题的贪婪随机平均块采样高斯-赛德尔（GRABGS）方法。GRABGS方法利用新颖的概率准则来收集控制索引坐标集，并通过在每次迭代中平均执行多次精确线搜索来最小化二次凸目标。概率准则旨在捕获范数相对较大的子向量。此外，GRABGS 方法被归类为随机块高斯-赛德尔方法的成员，可用于并行实现。收敛分析包含两种类型的外推步长：恒定步长和自适应步长。证明了当矩阵具有满列秩时，GRABGS方法收敛于稀疏线性最小二乘问题的唯一解。数值例子证明了该方法相对于贪婪随机坐标下降法和几种现有的最先进的块高斯-赛德尔方法的优越性。