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Existing results for low-rank matrix recovery largely focus on quadraticloss, which enjoys favorable properties such as restricted strongconvexity/smoothness (RSC/RSM) and well conditioning over all low rankmatrices. However, many interesting problems involve more general,non-quadratic losses, which do not satisfy such properties. For these problems,standard nonconvex approaches such as rank-constrained projected gradientdescent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorizationcould have poor empirical performance, and there is no satisfactory theoryguaranteeing global and fast convergence for these algorithms.  In this paper, we show that a critical component in provable low-rankrecovery with non-quadratic loss is a regularity projection oracle. This oraclerestricts iterates to low-rank matrices within an appropriate bounded set, overwhich the loss function is well behaved and satisfies a set of approximateRSC/RSM conditions. Accordingly, we analyze an (averaged) projected gradientmethod equipped with such an oracle, and prove that it converges globally andlinearly. Our results apply to a wide range of non-quadratic low-rankestimation problems including one bit matrix sensing/completion, individualizedrank aggregation, and more broadly generalized linear models with rankconstraints.

Low-rank matrix recovery with non-quadratic loss: projected gradient method and regularity projection oracle