Nonparametric estimation of low rank matrix valued function

Abstract

Let A: [0, 1] → Hm (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 E(Yj |τj, Xj) = 〈A(τj), Xj〉, where τj are i.i.d. uniformly distributed in [0, 1], Xj are i.i.d. matrix completion sampling matrices, Yj 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 L2-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∞-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 Lep-skii’s method and model selection with data splitting technique, which is computationally efficient and can be easily implemented and parallelized on distributed systems.