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The problem of structured matrix estimation has been studied mostly understrong noise dependence assumptions. This paper considers a general frameworkof noisy low-rank-plus-sparse matrix recovery, where the noise matrix may comefrom any joint distribution with arbitrary dependence across entries. Wepropose an incoherent-constrained least-square estimator and prove itstightness both in the sense of deterministic lower bound and matching minimaxrisks under various noise distributions. To attain this, we establish a novelresult asserting that the difference between two arbitrary low-rank incoherentmatrices must spread energy out across its entries, in other words cannot betoo sparse, which sheds light on the structure of incoherent low-rank matricesand may be of independent interest. We then showcase the applications of ourframework to several important statistical machine learning problems. In theproblem of estimating a structured Markov transition kernel, the proposedmethod achieves the minimax optimality and the result can be extended toestimating the conditional mean operator, a crucial component in reinforcementlearning. The applications to multitask regression and structured covarianceestimation are also presented. We propose an alternating minimization algorithmto approximately solve the potentially hard optimization problem. Numericalresults corroborate the effectiveness of our method which typically convergesin a few steps.

Structured Matrix Learning under Arbitrary Entrywise Dependence and Estimation of Markov Transition Kernel