We consider the nonconvex regularized method for low-rank matrix recovery. Under the assumption on the singular values of the parameter matrix, we provide the recovery bound for any stationary point of the nonconvex method by virtue of regularity conditions on the nonconvex loss function and the regularizer. This recovery bound can be much tighter than that of the convex nuclear norm regularized method when some of the singular values are larger than a threshold defined by the nonconvex regularizer. In addition, we consider the errors-in-variables matrix regression as an application of the nonconvex optimization method. Probabilistic consequences and the advantage of the nonoconvex method are demonstrated through verifying the regularity conditions for specific models with additive noise and missing data.

翻译：本文研究低秩矩阵恢复的非凸正则化方法。在参数矩阵奇异值的假设条件下，我们借助非凸损失函数与正则项的规则性条件，给出了该非凸方法任意稳定点的恢复误差界。当部分奇异值超过由非凸正则项定义的阈值时，该恢复界可显著优于凸核范数正则化方法的对应结果。此外，我们将非凸优化方法应用于含误差变量的矩阵回归问题。通过对具有加性噪声与缺失数据的特定模型验证规则性条件，论证了该方法的概率性结论及其相对于凸方法的优势。