The problem of structured matrix estimation has been studied mostly under strong noise dependence assumptions. This paper considers a general framework of noisy low-rank-plus-sparse matrix recovery, where the noise matrix may come from any joint distribution with arbitrary dependence across entries. We propose an incoherent-constrained least-square estimator and prove its tightness both in the sense of deterministic lower bound and matching minimax risks under various noise distributions. To attain this, we establish a novel result asserting that the difference between two arbitrary low-rank incoherent matrices must spread energy out across its entries, in other words cannot be too sparse, which sheds light on the structure of incoherent low-rank matrices and may be of independent interest. We then showcase the applications of our framework to several important statistical machine learning problems. In the problem of estimating a structured Markov transition kernel, the proposed method achieves the minimax optimality and the result can be extended to estimating the conditional mean operator, a crucial component in reinforcement learning. The applications to multitask regression and structured covariance estimation are also presented. We propose an alternating minimization algorithm to approximately solve the potentially hard optimization problem. Numerical results corroborate the effectiveness of our method which typically converges in a few steps.

翻译：结构化矩阵估计问题通常在强噪声依赖假设下进行研究。本文考虑一个通用的含噪低秩加稀疏矩阵恢复框架，其中噪声矩阵可来自任意联合分布且元素间具有任意依赖关系。我们提出一个非相干约束最小二乘估计器，并证明其在确定性下界及匹配多种噪声分布下极小化极大风险两方面的紧致性。为此，我们建立了一个新颖结论：任意两个低秩非相干矩阵的差值必然在其元素间分散能量，即不能过稀疏——这揭示了非相干低秩矩阵的结构特性，且可能具有独立研究价值。随后展示了该框架在若干重要统计机器学习问题中的应用。在结构化马尔可夫转移核估计问题中，所提方法达到极小化极大最优性，该结果可扩展至强化学习关键组件——条件均值算子的估计。同时给出了多任务回归和结构化协方差估计的应用案例。我们提出交替最小化算法以近似求解潜在的困难优化问题。数值结果证实了方法的有效性，通常数步内即可收敛。