Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. 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 gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing global and fast convergence for these algorithms. In this paper, we show that a critical component in provable low-rank recovery with non-quadratic loss is a regularity projection oracle. This oracle restricts iterates to low-rank matrices within an appropriate bounded set, over which the loss function is well behaved and satisfies a set of approximate RSC/RSM conditions. Accordingly, we analyze an (averaged) projected gradient method equipped with such an oracle, and prove that it converges globally and linearly. Our results apply to a wide range of non-quadratic low-rank estimation problems including one bit matrix sensing/completion, individualized rank aggregation, and more broadly generalized linear models with rank constraints.
翻译:低级别矩阵恢复的现有结果主要侧重于偏差损失,这种损失具有一些有利的特性,例如,强稳度/吸附性(RSC/RSM)有限,而且对所有低级别矩阵都有良好条件;然而,许多令人感兴趣的问题涉及更普遍的、非赤道性损失,无法满足这些特性;对于这些问题,标准的非凝固性方法,例如按等级限制预测的梯度下降(a.k.a.迭接硬阈值)和伯勒-蒙泰罗系数化等标准的非凝固性方法,其经验性能可能很差,而且没有令人满意的理论保证这些算法的全球和快速趋同。在本文件中,我们表明,在可实现的低级别与非夸大损失的低级别恢复中,有一个关键组成部分是规律性预测或触角。对于在适当约束性组合内,这种损失功能表现良好,满足了一套近乎RSC/RSM条件。 因此,我们分析了一种(平均)预测的梯度方法,配有这样的标志,并证明这种推合全球和线性损失的趋同性低级别,我们的结果被广泛地适用于一系列非普遍化的、低级的升级模型。