In this work, we first present an adaptive deterministic block coordinate descent method with momentum (mADBCD) to solve the linear least-squares problem, which is based on Polyak's heavy ball method and a new column selection criterion for a set of block-controlled indices defined by the Euclidean norm of the residual vector of the normal equation. The mADBCD method eliminates the need for pre-partitioning the column indexes of the coefficient matrix, and it also obviates the need to compute the Moore-Penrose pseudoinverse of a column sub-matrix at each iteration. Moreover, we demonstrate the adaptability and flexibility in the automatic selection and updating of the block control index set. When the coefficient matrix has full rank, the theoretical analysis of the mADBCD method indicates that it linearly converges towards the unique solution of the linear least-squares problem. Furthermore, by effectively integrating count sketch technology with the mADBCD method, we also propose a novel count sketch adaptive block coordinate descent method with momentum (CS-mADBCD) for solving highly overdetermined linear least-squares problems and analysis its convergence. Finally, numerical experiments illustrate the advantages of the proposed two methods in terms of both CPU times and iteration counts compared to recent block coordinate descent methods.

翻译：本文首先提出了一种自适应确定性块坐标下降动量方法（mADBCD）来求解线性最小二乘问题。该方法基于Polyak重球法，并结合一种新的列选择准则，该准则由法方程残差向量的欧几里得范数定义的一组块控制索引构成。mADBCD方法无需预先划分系数矩阵的列索引，也无需在每次迭代中计算列子矩阵的Moore-Penrose伪逆。此外，我们证明了该方法在块控制索引集的自动选择与更新方面具有适应性和灵活性。当系数矩阵满秩时，mADBCD方法的理论分析表明，它能以线性收敛速度逼近线性最小二乘问题的唯一解。进一步地，通过将计数素描技术有效整合到mADBCD方法中，我们还提出了一种新颖的计数素描自适应块坐标下降动量方法（CS-mADBCD），用于求解高度超定线性最小二乘问题，并分析了其收敛性。最后，数值实验表明，与近期的块坐标下降方法相比，所提出的两种方法在CPU时间和迭代次数方面均具有优势。