This paper concerns a class of low-rank composite factorization models arising from matrix completion. For this nonconvex and nonsmooth optimization problem, we propose a proximal alternating minimization algorithm (PAMA) with subspace correction, in which a subspace correction step is imposed on every proximal subproblem so as to guarantee that the corrected proximal subproblem has a closed-form solution. For this subspace correction PAMA, we prove the subsequence convergence of the iterate sequence, and establish the convergence of the whole iterate sequence and the column subspace sequences of factor pairs under the KL property of objective function and a restrictive condition that holds automatically for the column $\ell_{2,0}$-norm function. Numerical comparison with the proximal alternating linearized minimization method on one-bit matrix completion problems indicates that PAMA has an advantage in seeking lower relative error within less time.

翻译：本文研究一类源于矩阵补全的低秩复合分解模型。针对这一非凸非光滑优化问题，我们提出一种带子空间校正的邻近交替最小化算法（PAMA），该算法在每个邻近子问题上施加子空间校正步骤，以确保校正后的邻近子问题具有闭式解。对于此子空间校正PAMA，我们证明了迭代序列的子序列收敛性，并在目标函数满足KL性质及一个对列$\ell_{2,0}$-范数函数自动成立的限制条件下，建立了整个迭代序列以及因子对列子空间序列的收敛性。在单比特矩阵补全问题上与邻近交替线性化最小化方法的数值比较表明，PAMA在更短时间内获得更低相对误差方面具有优势。