  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 H\"{o}lder 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.
