Nonparametric Estimation of Low Rank Matrix Valued Function
This work addresses a fundamental problem in high-dimensional statistics for researchers in matrix estimation, though it appears incremental as it builds on existing methods like nuclear norm penalties and local polynomials.
The paper tackles the problem of statistically estimating a low-rank matrix-valued function from regression data, proposing nuclear norm penalized local polynomial and bias-reducing kernel estimators, and establishes optimal risk bounds up to logarithmic factors.
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 $Σ(β,L)$. The goal of this paper is to study statistical estimation of $A$ based on the regression model $\mathbb{E}(Y_j|τ_j,X_j) = \langle A(τ_j), X_j \rangle,$ where $τ_j$ 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.