广义加权低秩恢复的自适应秩延拓算法
An Adaptive Rank Continuation Algorithm for General Weighted Low-rank Recovery
摘要：
本文致力于提出一个通用的加权低秩恢复模型，并设计一个快速的无SVD计算方案来求解它。首先，我们的通用加权低秩恢复模型统一了文献中现有的几种方法。此外，我们的模型很容易扩展到非凸设置。在低秩恢复的文献中，大多数一阶近似算法都需要计算奇异值分解（SVD）。由于奇异值分解不随矩阵的维数适当地缩放，当问题规模变得更大时，这些算法变得更慢。通过将核范数的变分公式化到邻近梯度下降的子问题中，我们避免了计算SVD，从而显著提高了速度。此外，我们的算法保留了核范数[33]的秩识别性质，这进一步允许我们设计一个渐进实现最小迭代复杂度的秩连续方案。玩具的例子和现实世界的问题，包括结构从运动（SfM）和光度立体，背景估计，矩阵完成的数值实验，证明了我们所提出的算法的优越性。
英文摘要：
This paper is devoted to proposing a general weighted low-rank recovery model and designing a fast SVD-free computational scheme to solve it. First, our generic weighted low-rank recovery model unifies several existing approaches in the literature.~Moreover, our model readily extends to the non-convex setting. Algorithm-wise, most first-order proximal algorithms in the literature for low-rank recoveries require computing singular value decomposition (SVD). As SVD does not scale appropriately with the dimension of the matrices, these algorithms become slower when the problem size becomes larger. By incorporating the variational formulation of the nuclear norm into the sub-problem of proximal gradient descent, we avoid computing SVD, which results in significant speed-up. Moreover, our algorithm preserves the rank identification property of nuclear norm [33] which further allows us to design a rank continuation scheme that asymptotically achieves the minimal iteration complexity. Numerical experiments on both toy examples and real-world problems, including structure from motion (SfM) and photometric stereo, background estimation, and matrix completion, demonstrate the superiority of our proposed algorithm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
