In this paper, we propose {\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). The key step of RISRO is recursive importance sketching, a new sketching framework based on deterministically designed recursive projections, which significantly differs from the randomized sketching in the literature \citep{mahoney2011randomized,woodruff2014sketching}. Several existing algorithms in the literature can be reinterpreted under this 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 to solve a dimension-reduced least squares problem. We establish the local quadratic-linear and quadratic rate of convergence for RISRO under some mild conditions. We also discover a deep connection of RISRO to the Riemannian Gauss-Newton algorithm 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, 我们为 \ underline {R} ank} 提出限制的最小方程式 { underline} (RISRO) (RISRO) 。 RISRO 的关键步骤是重现重要素描, 一个新的素描框架基于确定性设计的循环预测, 这与文献\ citep{mahoney2011randomized, 木质鲁2014- ketching} 的随机草图有很大不同。 文献中的几种现有算法可以在新的素描框架和RISRO 提供明确的优势。 RISRO 很容易执行和计算效率, 每一次循环的核心程序是解决一个小维- 降最小方的问题。 我们建立了本地的矩- 线 和 RIRO 在某些温和条件下的合并率 New RIRO 的随机线和四重率率率 。 我们还在这个新的素描图框架下发现一个深度的高级性能和 IMIS 的轨迹学 。