基于同伦临近映射算法的低秩矩阵恢复

文章提出的同伦临近映射（homotopy proximity mapping，HPM）算法可用于从信号的（噪声）线性测量中重建低秩信号或从观测数据中学习低秩线性模型。该算法在每次迭代时采用核范数的简单临近映射，并逐渐减小核范数的正则化参数。结果表明，HPM算法可在有噪测量下进行低秩矩阵恢复，且恢复结果表现为全局线性收敛。此外，更大的观测值可使HPM算法恢复更准确、收敛更快。

In this paper, a homotopy proximity mapping (HPM) algorithm is proposed to reconstruct low-rank signals from noisy linear measurements of signals or to learn low-rank linear models from observed data. The algorithm adopts a simple proximity mapping of the kernel norm during each iteration, and gradually reduces the regularization parameters of the kernel norm. The experimental results show that HPM algorithm can perform low-rank matrix recovery under noisy measurements, and the recovery results exhibit global linear convergence. In addition, increasing observation values leads to not only more accurate recovery, but also faster convergence.