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Fisher Information and Support Complexity of Gaussian Mixtures — arXiv:2610.21726

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.21726
License (as archived): CC0-1.0
Archived: 2026-10-16T10:34:39.733Z
SHA-256 of the archived text: 1bd15e8bf314903ddb0b8c87206356eded1db381c7236be18e14860afa6a9d18
Stable URL of this page: https://metraipolis.com/documento/dc_fd730b08463d

How to cite

Fisher Information and Support Complexity of Gaussian Mixtures — arXiv:2610.21726 — arXiv (en), https://arxiv.org/abs/2610.21726 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_fd730b08463d (Metraipolis archive dc_fd730b08463d, sha256 1bd15e8bf314903ddb0b8c87206356eded1db381c7236be18e14860afa6a9d18)

BibTeX

@misc{dcfd730b08463d,
  title = {Fisher Information and Support Complexity of Gaussian Mixtures — arXiv:2610.21726},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.21726},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_fd730b08463d; sha256 1bd15e8bf314903ddb0b8c87206356eded1db381c7236be18e14860afa6a9d18},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.21726. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study lower bounds on the Fisher information of Gaussian location mixtures and their applications to estimation problems. First, for a discrete mixing distribution, we derive a universal positive-semidefinite lower bound on the Fisher information matrix in terms of the Rényi entropy of order two. For a mixing distribution supported on $K$ points, this bound scales as $\log(K)/K^2$, improving by a logarithmic factor on the support-size dependence obtained from Stam's inequality. We then develop dimension-aware trace bounds using classical Sobolev inequalities. These bounds are expressed in terms of Sibson's $\alpha$-mutual information and yield the scaling $K^{-2/n}$ in dimension $n$. A complementary bound involving the $\alpha$-mutual information of order $1/2$ is obtained using the sharp Young convolution inequality. The derived inequalities are further specialized to rotationally invariant mixing distributions supported on discrete spherical shells. In this setting, an intrinsic-dimensional bound on Sibson mutual information leads to lower bounds on the Fisher information in terms of the number of occupied shells. As applications, we study least favorable priors for bounded Gaussian mean estimation. In the scalar case, we show that the least favorable prior on $[-R,R]$ has at least $\Omega(R\sqrt{\log R})$ support points and that each of its probability masses is of order at most $1/R$. In the vector case, we show that the number of occupied shells grows at least linearly with $R$. Finally, Brown's identity and the I-MMSE relationship are used to translate the Fisher-information inequalities into upper bounds on the MMSE and mutual information.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.IT, nuovi lavori del giorno).
Categorie arXiv: Information Theory (cs.IT).
Pagina del lavoro: https://arxiv.org/abs/2610.21726 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T08:51:02.578Z dalla pagina https://arxiv.org/list/math.IT/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).