This paper studies the problem of recovering a low-rank matrix from several noisy random linear measurements. We consider the setting where the rank of the ground-truth matrix is unknown a priori and use an objective function built from a rank-overspecified factored representation of the matrix variable, where the global optimal solutions overfit and do not correspond to the underlying ground truth. We then solve the associated nonconvex problem using gradient descent with small random initialization. We show that as long as the measurement operators satisfy the restricted isometry property (RIP) with its rank parameter scaling with the rank of the ground-truth matrix rather than scaling with the overspecified matrix rank, gradient descent iterations are on a particular trajectory towards the ground-truth matrix and achieve nearly information-theoretically optimal recovery when it is stopped appropriately. We then propose an efficient stopping strategy based on the common hold-out method and show that it detects a nearly optimal estimator provably. Moreover, experiments show that the proposed validation approach can also be efficiently used for image restoration with deep image prior, which over-parameterizes an image with a deep network.

翻译：本文研究从若干含噪随机线性测量中恢复低秩矩阵的问题。我们考虑真实矩阵的秩先验未知的情形，并采用基于矩阵变量的秩过参数化分解表示构建的目标函数，其中全局最优解会过拟合且不对应于潜在的真实矩阵。随后，我们使用小规模随机初始化的梯度下降法求解相关非凸问题。研究表明，只要测量算子满足限制等距性（RIP）且其秩参数与真实矩阵的秩（而非过参数化矩阵的秩）成比例缩放，梯度下降迭代将沿着特定轨迹逼近真实矩阵，并在适时停止时实现近乎信息论最优的恢复效果。我们进一步提出基于常用留出法的高效停止策略，并证明该策略能够可验证地检测出近乎最优的估计量。此外，实验表明所提出的验证方法也可有效应用于基于深度图像先验的图像恢复任务，该方法通过深度网络对图像进行过参数化建模。