Kernel Mean Estimation via Spectral Filtering
The problem of estimating the kernel mean in a reproducing kernel Hilbert space (RKHS) is central to kernel methods in that it is used by classical approaches (e.g., when centering a kernel PCA matrix), and it also forms the core inference step of modern kernel methods (e.g., kernel-based non-parametric tests) that rely on embedding probability distributions in RKHSs. Previous work [1] has shown that shrinkage can help in constructing "better" estimators of the kernel mean than the empirical estimator. The present paper studies the consistency and admissibility of the estimators in [1], and proposes a wider class of shrinkage estimators that improve upon the empirical estimator by considering appropriate basis functions. Using the kernel PCA basis, we show that some of these estimators can be constructed using spectral filtering algorithms which are shown to be consistent under some technical assumptions. Our theoretical analysis also reveals a fundamental connection to the kernel-based supervised learning framework. The proposed estimators are simple to implement and perform well in practice.
This paper deals with the estimation of the kernel mean, an effective tool to design kernel-based learning algorithms for probability measures. Specifically, the authors propose the use of spectral filtering to obtain estimators of the kernel mean better than the standard empirical estimator. Their estimator builds upon recent work [13] which demonstrates  that the commonly used kernel mean estimator can be improved by shrinkage estimators. The authors provide a theoretical analysis of their spectral kernel mean estimator from an operator learning point of view and conducted artificial and real data experiments to test their spectral-based mean estimation under various settings and compare it with the standard empirical estimator and other shrinkage estimators.The paper is well written and the subject is well motivated in the introduction. The paper clearly explains a new mechanism that should have margin to improve kernel mean estimation. The underlying idea is quite simple, but it is proved through a rigorous analysis and an empirical evaluation.In the experimental section, it is true that Figure 2 shows that  the spectral estimator is less expensive than other shrinkage estimator in terms of computational cost; however, it is much more expensive than the standard estimator. The data used in the experiments seem to be relatively small. The computational efficiency of the proposed estimation for large data is not clear.Minor comment: - line 87: better than \mu —> better than \hat{\mu}- line 142: There is not (iii) in theorem 1Pros: 	- new kernel mean estimator         - well written paper         - the proposed method is justified by a theoretical analysis and experimental resultsCons: 	- The data used in the experiments seem to be relatively small