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Dimension-free comparison inequalities between f-divergences for Gaussian location mixtures

  • Jeonghwan Lee
  • Cong Ma
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Abstract

We establish dimension-free comparison inequalities between the Kullback-Leibler (KL) and χ2-divergences and the squared Hellinger distance for Gaussian location mixtures with compactly supported mixing distributions. We show that both divergences are bounded by the squared Hellinger distance up to constants depending only on the radius of the support, independently of the ambient dimension and the number of mixture components. These results resolve an open question raised by \cite{jia2023entropic} and strengthen existing comparison inequalities. Our proof relies on a Gaussian interpolation argument that relates the derivatives of these divergences to differences between posterior means.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026dimensionfree,
  title = {{Dimension-free comparison inequalities between $f$-divergences for Gaussian location mixtures}},
  author = {Jeonghwan Lee and Cong Ma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25070},
  eprint = {2610.25070},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25070},
}
RIS
TY  - UNPB
TI  - Dimension-free comparison inequalities between f-divergences for Gaussian location mixtures
AU  - Jeonghwan Lee
AU  - Cong Ma
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25070
UR  - https://arxiv.org/abs/2610.25070
AB  - We establish dimension-free comparison inequalities between the Kullback-Leibler (KL) and χ^2-divergences and the squared Hellinger distance for Gaussian location mixtures with compactly supported mixing distributions. We show that both divergences are bounded by the squared Hellinger distance up to constants depending only on the radius of the support, independently of the ambient dimension and the number of mixture components. These results resolve an open question raised by \cite{jia2023entropic} and strengthen existing comparison inequalities. Our proof relies on a Gaussian interpolation argument that relates the derivatives of these divergences to differences between posterior means.
ER  -