In this paper, we propose a new {\it \underline{R}ecursive} {\it \underline{I}mportance} {\it \underline{S}ketching} algorithm for {\it \underline{R}ank} constrained least squares {\it \underline{O}ptimization} (RISRO). As its name suggests, the algorithm is based on a new sketching framework, recursive importance sketching. Several existing algorithms in the literature can be reinterpreted under the new sketching framework and RISRO offers clear advantages over them. RISRO is easy to implement and computationally efficient, where the core procedure in each iteration is only solving a dimension reduced least squares problem. Different from numerous existing algorithms with locally geometric convergence rate, we establish the local quadratic-linear and quadratic rate of convergence for RISRO under some mild conditions. In addition, we discover a deep connection of RISRO to Riemannian manifold optimization on fixed rank matrices. The effectiveness of RISRO is demonstrated in two applications in machine learning and statistics: low-rank matrix trace regression and phase retrieval. Simulation studies demonstrate the superior numerical performance of RISRO.

翻译：在本文中, 我们提出一个新的 \ sunderline { R} 线下 { R} 直线} 。 文献中的一些现有算法可以在新的素描框架下重新解释, RISRO提供了明确的优势。 RISRO很容易执行和计算效率, 在每个迭代中的核心程序只能解决一个减少最小平方问题的维度。 不同于与本地几何趋同率相关的多种现有算法, 我们为RISRO在一些温和条件下建立了本地的二次线性与二次趋同率。 此外, 我们发现, RISRO 和 Riemann 的低级矩阵的多重优化有着深层次的联系。 RISRO 的实效性能和计算效率很容易实现。 两次高级的 IMIS 模型研究展示了两次高级的回溯性研究 。