Nonparametric Estimation of Low Rank Matrix Valued Function

Let $A:[0,1]\rightarrow\mathbb{H}_m$ (the space of Hermitian matrices) be a matrix valued function which is low rank with entries in Hölder class $\Sigma(\beta,L)$. The goal of this paper is to study statistical estimation of $A$ based on the regression model $\mathbb{E}(Y_j|\tau_j,X_j) = \langle A(\tau_j), X_j \rangle,$ where $\tau_j$ are i.i.d. uniformly distributed in $[0,1]$, $X_j$ are i.i.d. matrix completion sampling matrices, $Y_j$ are independent bounded responses. We propose an innovative nuclear norm penalized local polynomial estimator and establish an upper bound on its point-wise risk measured by Frobenius norm. Then we extend this estimator globally and prove an upper bound on its integrated risk measured by $L_2$-norm. We also propose another new estimator based on bias-reducing kernels to study the case when $A$ is not necessarily low rank and establish an upper bound on its risk measured by $L_{\infty}$-norm. We show that the obtained rates are all optimal up to some logarithmic factor in minimax sense. Finally, we propose an adaptive estimation procedure based on Lepskii's method and model selection with data splitting which is computationally efficient and can be easily implemented and parallelized.