Structured Matrix Learning under Arbitrary Entrywise Dependence and Estimation of Markov Transition Kernel
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任意逐元素依赖下的结构矩阵学习与马尔可夫转移核估计
- 研究问题
- 当噪声矩阵元素之间可以任意依赖时，如何恢复低秩加稀疏结构，并将估计方法用于结构化马尔可夫转移核？
- 方法与思路
- 作者提出非相干约束最小二乘估计，分析低秩非相干矩阵之差的能量分散性质，并用交替最小化近似求解。
- 材料陈述的贡献
- 摘要称建立确定性下界及多类噪声下匹配的极小极大风险，转移核应用达到最优性，并扩展至条件均值算子等问题。
- 阅读关注点
- 应核对任意依赖与风险界所需的噪声条件、低秩和稀疏分量的可区分性，以及近似算法误差与统计保证的联系。
- 阅读边界
- 所给版本为 arXiv 预印本，不能据此认定已同行评审或被期刊接收；摘要未给出各噪声类的风险界及算法收敛条件。
原文摘要
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, it 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.