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