论文标题
多响应回归中的低率矩阵估计带有测量误差：统计和计算保证
Low-rank matrix estimation in multi-response regression with measurement errors: Statistical and computational guarantees
论文作者
论文摘要
在本文中，我们研究了具有测量误差的多响应回归模型中的矩阵估计问题。提出了基于修正的损耗函数和核标准正常器的组合的非convex误差校正器，以估计矩阵参数。然后，在（近）低级别假设下，我们从一般的角度分析了非凸正则化估计器的统计和计算理论特性。在统计方面，我们建立了针对非凸估计量的任何全局解决方案的非反应恢复，在损失函数的强凸度受限的情况下。在计算方面，我们通过近端梯度方法解决了非convex优化问题。该算法被证明是收敛到几乎全球溶液并达到线性收敛速率的。此外，我们还验证了足够的条件以保持一般结果，以便为特定类型的测量误差（包括加性噪声和缺少数据）获得概率后果。最后，通过几个数值实验在损坏的多变量多响应回归模型上证明了理论后果。仿真结果揭示了在高维缩放下与我们的理论一致性的一致性。
In this paper, we investigate the matrix estimation problem in the multi-response regression model with measurement errors. A nonconvex error-corrected estimator based on a combination of the amended loss function and the nuclear norm regularizer is proposed to estimate the matrix parameter. Then under the (near) low-rank assumption, we analyse statistical and computational theoretical properties of global solutions of the nonconvex regularized estimator from a general point of view. In the statistical aspect, we establish the nonasymptotic recovery bound for any global solution of the nonconvex estimator, under restricted strong convexity on the loss function. In the computational aspect, we solve the nonconvex optimization problem via the proximal gradient method. The algorithm is proved to converge to a near-global solution and achieve a linear convergence rate. In addition, we also verify sufficient conditions for the general results to be held, in order to obtain probabilistic consequences for specific types of measurement errors, including the additive noise and missing data. Finally, theoretical consequences are demonstrated by several numerical experiments on corrupted errors-in-variables multi-response regression models. Simulation results reveal excellent consistency with our theory under high-dimensional scaling.
