Motivated by the increasing use of kernel-based metrics for high-dimensional and large-scale data, we study the asymptotic behaviour of kernel two-sample tests when the dimension and sample sizes both diverge to infinity. We focus on the maximum mean discrepancy using an isotropic kernel, which includes maximum mean discrepancy with the Gaussian kernel and the Laplace kernel, and the energy distance as special cases. We derive asymptotic expansions of the kernel two-sample statistics, based on which we establish a central limit theorem under both the null hypothesis and the local and fixed alternatives. The new nonnull central limit theorem results allow us to perform asymptotic exact power analysis, which reveals a delicate interplay between the moment discrepancy that can be detected by the kernel two-sample tests and the dimension-and-sample orders. The asymptotic theory is further corroborated through numerical studies.
Finding a suitable representation of multivariate data is fundamental in many scientific disciplines. Projection pursuit ( ) aims to extract interesting ‘non‐Gaussian’ features from multivariate data, and tends to be computationally intensive even when applied to data of low dimension. In high‐dimensional settings, a recent work (Bickel et al., 2018) on addresses asymptotic characterization and conjectures of the feasible projections as the dimension grows with sample size. To gain practical utility of and learn theoretical insights into in an integral way, data analytic tools needed to evaluate the behaviour of in high dimensions become increasingly desirable but are less explored in the literature. This paper focuses on developing computationally fast and effective approaches central to finite sample studies for (i) visualizing the feasibility of in extracting features from high‐dimensional data, as compared with alternative methods like and , and (ii) assessing the plausibility of in cases where asymptotic studies are lacking or unavailable, with the goal of better understanding the practicality, limitation and challenge of in the analysis of large data sets.
more » « less- NSF-PAR ID:
- 10405192
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- International Statistical Review
- Volume:
- 91
- Issue:
- 1
- ISSN:
- 0306-7734
- Page Range / eLocation ID:
- p. 140-161
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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