Estimation of Low-Rank Covariance Function

TL;DR

This paper introduces a new nuclear norm penalization method for estimating low-rank covariance functions of Gaussian processes, achieving optimality and outperforming traditional sample covariance estimators.

Contribution

The paper proposes a novel adaptive estimation procedure that simultaneously exploits low-rank and smoothness properties of covariance functions, with proven optimality and improved performance.

Findings

Method outperforms sample covariance by polynomial factor in n

Achieves minimax optimality for low-rank covariance estimation

Provides a scheme to estimate the noise variance

Abstract

We consider the problem of estimating a low rank covariance function $K(t,u)$ of a Gaussian process $S(t), t\in [0,1]$ based on $n$ i.i.d. copies of $S$ observed in a white noise. We suggest a new estimation procedure adapting simultaneously to the low rank structure and the smoothness of the covariance function. The new procedure is based on nuclear norm penalization and exhibits superior performances as compared to the sample covariance function by a polynomial factor in the sample size $n$. Other results include a minimax lower bound for estimation of low-rank covariance functions showing that our procedure is optimal as well as a scheme to estimate the unknown noise variance of the Gaussian process.