Cluster-Seeking James-Stein Estimators

TL;DR

This paper introduces adaptive clustering-based shrinkage estimators for high-dimensional Gaussian mean estimation, which infer the underlying structure of the parameter vector to achieve significant risk reduction.

Contribution

The paper proposes a novel data-driven clustering approach to construct shrinkage estimators that adaptively infer the structure of the parameter vector for improved estimation.

Findings

Estimators achieve significant risk reduction over ML for large dimensions.

Theoretical concentration and convergence results support the effectiveness of the estimators.

Simulation results validate the theoretical advantages of the proposed methods.

Abstract

This paper considers the problem of estimating a high-dimensional vector of parameters $\boldsymbol{\theta} \in \mathbb{R}^n$ from a noisy observation. The noise vector is i.i.d. Gaussian with known variance. For a squared-error loss function, the James-Stein (JS) estimator is known to dominate the simple maximum-likelihood (ML) estimator when the dimension $n$ exceeds two. The JS-estimator shrinks the observed vector towards the origin, and the risk reduction over the ML-estimator is greatest for $\boldsymbol{\theta}$ that lie close to the origin. JS-estimators can be generalized to shrink the data towards any target subspace. Such estimators also dominate the ML-estimator, but the risk reduction is significant only when $\boldsymbol{\theta}$ lies close to the subspace. This leads to the question: in the absence of prior information about $\boldsymbol{\theta}$, how do we design estimators that give significant risk reduction over the ML-estimator for a wide range of $\boldsymbol{\theta}$? In this paper, we propose shrinkage estimators that attempt to infer the structure of $\boldsymbol{\theta}$ from the observed data in order to construct a good attracting subspace. In particular, the components of the observed vector are separated into clusters, and the elements in each cluster shrunk towards a common attractor. The number of clusters and the attractor for each cluster are determined from the observed vector. We provide concentration results for the squared-error loss and convergence results for the risk of the proposed estimators. The results show that the estimators give significant risk reduction over the ML-estimator for a wide range of $\boldsymbol{\theta}$, particularly for large $n$. Simulation results are provided to support the theoretical claims.