Kernel Mean Estimation and Stein's Effect
This work addresses a fundamental issue in kernel methods for machine learning, offering incremental improvements to kernel mean estimation.
The paper tackles the problem of estimating kernel means from finite samples and shows that the standard empirical average estimator can be improved using Stein's phenomenon, resulting in shrinkage estimators that outperform it in benchmark applications.
A mean function in reproducing kernel Hilbert space, or a kernel mean, is an important part of many applications ranging from kernel principal component analysis to Hilbert-space embedding of distributions. Given finite samples, an empirical average is the standard estimate for the true kernel mean. We show that this estimator can be improved via a well-known phenomenon in statistics called Stein's phenomenon. After consideration, our theoretical analysis reveals the existence of a wide class of estimators that are better than the standard. Focusing on a subset of this class, we propose efficient shrinkage estimators for the kernel mean. Empirical evaluations on several benchmark applications clearly demonstrate that the proposed estimators outperform the standard kernel mean estimator.