Nonparametric Estimation of Low Rank Matrix Valued Function

TL;DR

This paper develops and analyzes new nonparametric estimators for low rank Hermitian matrix-valued functions, providing optimal risk bounds and an adaptive, computationally efficient procedure for practical implementation.

Contribution

It introduces a nuclear norm penalized local polynomial estimator and a bias-reducing kernel estimator with proven optimal rates, extending nonparametric matrix function estimation to the low rank setting.

Findings

Proposed estimators achieve near-minimax optimal risk bounds.

Extended estimators to the non-low-rank case with risk bounds.

Developed an adaptive, computationally efficient estimation procedure.

Abstract

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.