The kernel mean embedding of a random probability distribution and its principal components
Abstract
Given a random probability distribution F, whose realizations are defined on ℝd, and a kernel k, our aim is to analyze the properties of the kernel mean embedding, μF, of F as a random element of the corresponding reproducing kernel Hilbert space ℋ. This establishes a rigorous framework in which kernel statistical techniques for distributional data can be stated both in the sample and population settings and, thus, the asymptotic properties of sample procedures can be properly established. As an illustration of this approach we focus on the principal components of F. We obtain a general convergence result and the explicit expression of the spectral decomposition for the uncentered covariance operator of a normal probability distribution with a Gaussian distributed expectation. The performance of this kernel principal components procedure is illustrated with simulations and the analysis of the intensity distribution in synthetic aperture radar images.
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BibTeX
@misc{baillo2026kernel,
title = {{The kernel mean embedding of a random probability distribution and its principal components}},
author = {Amparo Baíllo and Martín Sánchez-Signorini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33286},
eprint = {2610.33286},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33286},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The kernel mean embedding of a random probability distribution and its principal components
AU - Amparo Baíllo
AU - Martín Sánchez-Signorini
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33286
UR - https://arxiv.org/abs/2610.33286
AB - Given a random probability distribution F, whose realizations are defined on ℝ^d, and a kernel k, our aim is to analyze the properties of the kernel mean embedding, μ_F, of F as a random element of the corresponding reproducing kernel Hilbert space ℋ. This establishes a rigorous framework in which kernel statistical techniques for distributional data can be stated both in the sample and population settings and, thus, the asymptotic properties of sample procedures can be properly established. As an illustration of this approach we focus on the principal components of F. We obtain a general convergence result and the explicit expression of the spectral decomposition for the uncentered covariance operator of a normal probability distribution with a Gaussian distributed expectation. The performance of this kernel principal components procedure is illustrated with simulations and the analysis of the intensity distribution in synthetic aperture radar images.
ER -