Admissible kernels for RKHS embedding of probability distributions

Liangzhi Chen (), Thomas Hotz () and Haizhang Zhang ()
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Liangzhi Chen: Sun Yet-sen University
Thomas Hotz: Institute of Mathematics
Haizhang Zhang: Sun Yat-sen University

Statistical Papers, 2021, vol. 62, issue 3, No 18, 1499-1518

Abstract: Abstract Similarity measurement of two probability distributions is important in many applications of statistics. Embedding such distributions into a reproducing kernel Hilbert space (RKHS) has many favorable properties. The choice of the reproducing kernel is crucial in the approach. We study this question by considering the similarity of two distributions of the same class. In particular, we investigate when the RKHS embedding is “admissible” in the sense that the distance between the embeddings should become smaller when the expectations are getting closer or when the variance is increasing to infinity. We give conditions on the widely-used translation-invariant reproducing kernels to be admissible. We also extend the study to multivariate non-symmetric Gaussian distributions.

Keywords: Gaussian distributions; Reproducing kernels; RKHS embedding; Translation-invariant kernels; Radially decreasing functions (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s00362-019-01144-5

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