We study deterministic matrix completion problem, i.e., recovering a low-rank matrix from a few observed entries where the sampling set is chosen as the edge set of a Ramanujan graph. We first investigate projected gradient descent (PGD) applied to a Burer-Monteiro least-squares problem and show that it converges linearly to the incoherent ground-truth with respect to the condition number \k{appa} of ground-truth under a benign initialization and large samples. We next apply the scaled variant of PGD to deal with the ill-conditioned case when \k{appa} is large, and we show the algorithm converges at a linear rate independent of the condition number \k{appa} under similar conditions. Finally, we provide numerical experiments to corroborate our results.

翻译：我们研究确定性矩阵补全问题，即从少量观测条目中恢复低秩矩阵，其中采样集选用拉马努金图的边集。首先探究应用于Burer-Monteiro最小二乘问题的投影梯度下降（PGD）方法，证明在良性初始化及大样本条件下，该方法相对于真实矩阵的条件数\k{appa}呈线性收敛至非相干真实解。其次采用PGD的缩放变体处理\k{appa}较大时的病态情形，证实在类似条件下该算法以独立于条件数\k{appa}的线性速率收敛。最后通过数值实验验证理论结果。