Kernel Mean Estimation and Stein Effect

A mean function in reproducing kernel Hilbert space (RKHS), or a kernel mean, is an important part of many algorithms ranging from kernel principal component analysis to Hilbert-space embedding of distributions. Given a finite sample, an empirical average is the standard estimate for the true kernel mean. We show that this estimator can be improved due to a well-known phenomenon in statistics called Stein phenomenon. After consideration, our theoretical analysis reveals the existence of a wide class of estimators that are better than the standard one. Focusing on a subset of this class, we propose efficient shrinkage estimators for the kernel mean. Empirical evaluations on several applications clearly demonstrate that the proposed estimators outperform the standard kernel mean estimator.