论文标题
通过Unlifted凸优化从排名一的投影中估算低级别矩阵
Low-Rank Matrix Estimation From Rank-One Projections by Unlifted Convex Optimization
论文作者
论文摘要
我们研究具有凸式公式的估计器，用于从排名一的投影中恢复低级矩阵。使用对目标$ d_1 \ times d_2 $ rank-$ r $矩阵的初始估计，估算器承认在尺寸$ r（d_1+d_2）$的空间中运行的实用亚级别方法。该属性使估计器比基于提升和半决赛编程的凸估计器更明显地扩展。此外，我们提出了在实际高斯测量模型下进行精确恢复的简化分析，以及使用球形$ t $ -DESIGN的部分偏差测量模型。我们表明，在两个模型下，估计器都成功，如果测量次数超过$ r^2（d_1+d_2）$，则达到某些对数因素。该样本复杂性改善了非概念迭代算法的现有结果。
We study an estimator with a convex formulation for recovery of low-rank matrices from rank-one projections. Using initial estimates of the factors of the target $d_1\times d_2$ matrix of rank-$r$, the estimator admits a practical subgradient method operating in a space of dimension $r(d_1+d_2)$. This property makes the estimator significantly more scalable than the convex estimators based on lifting and semidefinite programming. Furthermore, we present a streamlined analysis for exact recovery under the real Gaussian measurement model, as well as the partially derandomized measurement model by using the spherical $t$-design. We show that under both models the estimator succeeds, with high probability, if the number of measurements exceeds $r^2 (d_1+d_2)$ up to some logarithmic factors. This sample complexity improves on the existing results for nonconvex iterative algorithms.
