Rank minimization can be converted into tractable surrogate problems, such as Nuclear Norm Minimization (NNM) and Weighted NNM (WNNM). The problems related to NNM, or WNNM, can be solved iteratively by applying a closed-form proximal operator, called Singular Value Thresholding (SVT), or Weighted SVT, but they suffer from high computational cost of Singular Value Decomposition (SVD) at each iteration. We propose a fast and accurate approximation method for SVT, that we call fast randomized SVT (FRSVT), with which we avoid direct computation of SVD. The key idea is to extract an approximate basis for the range of the matrix from its compressed matrix. Given the basis, we compute partial singular values of the original matrix from the small factored matrix. In addition, by developping a range propagation method, our method further speeds up the extraction of approximate basis at each iteration. Our theoretical analysis shows the relationship between the approximation bound of SVD and its effect to NNM via SVT. Along with the analysis, our empirical results quantitatively and qualitatively show that our approximation rarely harms the convergence of the host algorithms. We assess the efficiency and accuracy of the proposed method on various computer vision problems, e.g., subspace clustering, weather artifact removal, and simultaneous multi-image alignment and rectification.

中文翻译：

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快速随机奇异值阈值化，用于低等级优化

等级最小化可以转换为易于处理的替代问题，例如核规范最小化（NNM）和加权NNM（WNNM）。可以通过应用称为奇异值阈值（SVT）或加权SVT的闭式近端算子来迭代地解决与NNM或WNNM有关的问题，但它们每次都遭受奇异值分解（SVD）的高计算成本的困扰迭代。我们提出了一种用于SVT的快速且准确的近似方法，我们将其称为快速随机SVT（FRSVT），从而避免了SVD的直接计算。关键思想是从其压缩矩阵中提取矩阵范围的近似基础。给定基础，我们从小因子矩阵计算原始矩阵的部分奇异值。此外，通过开发范围传播方法，我们的方法进一步加快了每次迭代时近似基的提取。我们的理论分析显示了SVD的近似边界与其通过SVT对NNM的影响之间的关系。与分析一起，我们的经验结果定量和定性地表明，我们的近似值很少损害主机算法的收敛性。我们评估了所提出的方法在各种计算机视觉问题上的效率和准确性，例如，子空间聚类，天气伪影去除以及同时的多图像对齐和校正。我们的经验结果定量和定性地表明，我们的近似值很少损害主机算法的收敛性。我们评估了所提出的方法在各种计算机视觉问题上的效率和准确性，例如，子空间聚类，天气伪影去除以及同时的多图像对齐和校正。我们的经验结果定量和定性地表明，我们的近似值很少损害主机算法的收敛性。我们评估了所提出的方法在各种计算机视觉问题上的效率和准确性，例如，子空间聚类，天气伪影去除以及同时的多图像对齐和校正。