This paper focuses on the convergence certificates of the majorized proximal alternating minimization (PAM) method with subspace correction, proposed in \cite{TaoQianPan22} for the column $\ell_{2,0}$-norm regularized factorization model and now extended to a class of low-rank composite factorization models from matrix completion. The convergence analysis of this PAM method becomes extremely challenging because a subspace correction step is introduced to every proximal subproblem to ensure a closed-form solution. We establish the full convergence of the iterate sequence and column subspace sequences of factor pairs generated by the PAM, under the KL property of the objective function and a condition that holds automatically for the column $\ell_{2,0}$-norm function. Numerical comparison with the popular proximal alternating linearized minimization (PALM) method is conducted on one-bit matrix completion problems, which indicates that the PAM with subspace correction has an advantage in seeking lower relative error within less time.

翻译：本文聚焦于具有子空间校正的主化邻近交替最小化方法的收敛性证明，该方法最初在文献\cite{TaoQianPan22}中针对列$\ell_{2,0}$-范数正则化分解模型提出，现被推广至一类源自矩阵补全的低秩复合分解模型。该PAM方法的收敛性分析变得极具挑战性，因为为确保闭式解，在每个邻近子问题中都引入了子空间校正步骤。在目标函数满足KL性质以及一个对列$\ell_{2,0}$-范数函数自动成立的条件之下，我们建立了由PAM生成的迭代序列及因子对列子空间序列的完全收敛性。在一比特矩阵补全问题上与流行的邻近交替线性化最小化方法进行了数值比较，结果表明具有子空间校正的PAM方法在更短时间内寻求更低相对误差方面具有优势。