SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1119-1147, January 2021.
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.

中文翻译：

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基于未提升凸优化的秩一投影的低秩矩阵估计

SIAM 矩阵分析与应用杂志，第 42 卷，第 3 期，第 1119-1147 页，2021 年 1 月。
我们研究了一个带有凸公式的估计器，用于从一阶投影中恢复低阶矩阵。使用 rank-$r$ 的目标 $d_1\times d_2$ 矩阵的因子的初始估计，估计器承认在维度为 $r(d_1+d_2)$ 的空间中操作的实用次梯度方法。这个属性使得估计器比基于提升和半定规划的凸估计器更具可扩展性。此外，我们在真实高斯测量模型以及使用球形 $t$-design 的部分去随机测量模型下对精确恢复进行了简化分析。我们表明，在两种模型下，如果测量次数超过 $r^2 (d_1+d_2)$ 达到某些对数因子，则估计器成功的概率很高。