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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 newsketching framework based on deterministically designed recursive projections,which significantly differs from the randomized sketching in the literature\citep{mahoney2011randomized,woodruff2014sketching}. Several existingalgorithms in the literature can be reinterpreted under this new sketchingframework and RISRO offers clear advantages over them. RISRO is easy toimplement and computationally efficient, where the core procedure in eachiteration is to solve a dimension-reduced least squares problem. We establishthe local quadratic-linear and quadratic rate of convergence for RISRO undersome mild conditions. We also discover a deep connection of RISRO to theRiemannian Gauss-Newton algorithm on fixed rank matrices. The effectiveness ofRISRO is demonstrated in two applications in machine learning and statistics:low-rank matrix trace regression and phase retrieval. Simulation studiesdemonstrate the superior numerical performance of RISRO.

Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-order Convergence