Statistical Bergman geometry
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913049010503680 |
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| author | Cho, Gunhee Yum, Jihun |
| author_facet | Cho, Gunhee Yum, Jihun |
| contents | This paper explores the Bergman geometry of bounded domains $Ω$ in $\mathbb{C}^n$ through the lens of information geometry by introducing a mapping $Φ: Ω\rightarrow \mathcal{P}(Ω)$, where $\mathcal{P}(Ω)$ denotes a space of probability measures on $Ω$. A result by J. Burbea and C. Rao establishes that the pullback of the Fisher information metric, the fundamental Riemannian pseudo-metric in information geometry, via $Φ$ coincides with the Bergman metric of $Ω$. Building on this idea, we consider $Ω$ as a statistical model and present several interesting results within this framework.
First, we derive a new statistical curvature formula for the Bergman metric by expressing it in terms of covariance. Second, given a proper holomorphic map $f: Ω_1 \rightarrow Ω_2$, we prove that if the induced measure push-forward $κ: \mathcal{P}(Ω_1) \rightarrow \mathcal{P}(Ω_2)$ preserves the Fisher information metrics, then $f$ must be a biholomorphism. Finally, we establish the consistency and the central limit theorem of the Fréchet sample mean for Calabi's diastasis function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_10207 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Statistical Bergman geometry Cho, Gunhee Yum, Jihun Complex Variables Differential Geometry This paper explores the Bergman geometry of bounded domains $Ω$ in $\mathbb{C}^n$ through the lens of information geometry by introducing a mapping $Φ: Ω\rightarrow \mathcal{P}(Ω)$, where $\mathcal{P}(Ω)$ denotes a space of probability measures on $Ω$. A result by J. Burbea and C. Rao establishes that the pullback of the Fisher information metric, the fundamental Riemannian pseudo-metric in information geometry, via $Φ$ coincides with the Bergman metric of $Ω$. Building on this idea, we consider $Ω$ as a statistical model and present several interesting results within this framework. First, we derive a new statistical curvature formula for the Bergman metric by expressing it in terms of covariance. Second, given a proper holomorphic map $f: Ω_1 \rightarrow Ω_2$, we prove that if the induced measure push-forward $κ: \mathcal{P}(Ω_1) \rightarrow \mathcal{P}(Ω_2)$ preserves the Fisher information metrics, then $f$ must be a biholomorphism. Finally, we establish the consistency and the central limit theorem of the Fréchet sample mean for Calabi's diastasis function. |
| title | Statistical Bergman geometry |
| topic | Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2305.10207 |