This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian matrix of the loss function, we establish an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, we establish its KL property of exponent 1/2 on the global optimal solution set under the noisy and full sample setting, and achieve this property at its certain class of critical points under the noisy and partial sample setting. These theoretical findings are also confirmed by solving the squared F-norm regularized factorization problem with an accelerated alternating minimization method.

本文关注的是用于噪声低秩矩阵恢复问题的平方 F(robenius)-范数正则化分解形式。在对损失函数的 Hessian 矩阵的限制条件数进行适当假设的情况下，对于秩不大于真矩阵的非严格临界点，我们建立了真矩阵的误差界。然后，对于平方F范数正则化因式分解最小二乘损失函数，我们在噪声和全样本设置下的全局最优解集上建立其指数为1/2的KL特性，并在其某类临界点上实现该特性在嘈杂和部分样本设置下。这些理论发现也通过使用加速交替最小化方法解决平方 F 范数正则化分解问题得到证实。