论文标题
验证方法过度参数化矩阵和图像恢复
A Validation Approach to Over-parameterized Matrix and Image Recovery
论文作者
论文摘要
本文研究了从几个嘈杂的随机线性测量结果中恢复低级别基质的问题。我们考虑以下设置的设置，即基地矩阵的等级是未知的，并且使用基于矩阵变量的排名量规定的分组表示的目标函数，其中全局最佳解决方案过度拟合，并且与基本的地面真理不符。然后，我们使用梯度下降和小的随机初始化解决了相关的非凸问题。我们表明，只要测量操作员以其等级参数缩放的范围来满足受限制的等轴测属性（RIP），而不是地面真相矩阵的等级，而不是按照过度指定的矩阵等级进行缩放，那么梯度下降迭代就可以在特定的轨迹上朝着基本矩阵并在理论上取得了几乎是最佳恢复的特定轨迹。然后，我们提出了一种基于共同持有方法的有效停止策略，并表明它可以检测到一个几乎最佳的估计器。此外，实验表明，所提出的验证方法也可以有效地用于图像恢复，并具有深层图像先验，该图像过度参与了使用深网络的图像。
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.
