Beyond Dual Flatness: Curvature Emergence via Anisotropic Metric Perturbations
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2025
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| _version_ | 1866901097815212032 |
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| author | HIDEKI |
| author_facet | HIDEKI |
| contents | <p>Dually flat manifolds occupy a privileged position in information geometry, enabling closed-form geodesics, canonical divergences, and the celebrated Pythagorean theorem for Bregman divergences. Yet the structural conditions that preserve or destroy this flatness under metric perturbation remain incompletely characterized. We introduce a dichotomy framework distinguishing two classes of perturbation: isotropic perturbations, which uniformly scale the base divergence, and anisotropic point-dependent perturbations, which introduce directional selectivity modulated by position in parameter space. Our main result establishes that this dichotomy precisely characterizes the stability of the Pythagorean theorem: isotropic perturbations preserve the theorem for arbitrary perturbation magnitude, while anisotropic point-dependent perturbations generically violate it. The key mechanism is that point-dependent modulation induces non-vanishing third-order mixed derivatives, which in turn generate curvature in the dual connections. Numerical experiments on 2D Gaussian (ε_max = 9.75) and 4-category categorical (ε_max = 0.435) manifolds confirm the theoretical predictions with machine-precision agreement. These results provide a minimal sufficient condition for curvature emergence and offer practical guidance for the design of geometrically faithful divergence modifications.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18019235 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Beyond Dual Flatness: Curvature Emergence via Anisotropic Metric Perturbations HIDEKI information geometry dual flatness Pythagorean theorem metric perturbation dual connections curvature emergence Bregman divergence statistical manifolds <p>Dually flat manifolds occupy a privileged position in information geometry, enabling closed-form geodesics, canonical divergences, and the celebrated Pythagorean theorem for Bregman divergences. Yet the structural conditions that preserve or destroy this flatness under metric perturbation remain incompletely characterized. We introduce a dichotomy framework distinguishing two classes of perturbation: isotropic perturbations, which uniformly scale the base divergence, and anisotropic point-dependent perturbations, which introduce directional selectivity modulated by position in parameter space. Our main result establishes that this dichotomy precisely characterizes the stability of the Pythagorean theorem: isotropic perturbations preserve the theorem for arbitrary perturbation magnitude, while anisotropic point-dependent perturbations generically violate it. The key mechanism is that point-dependent modulation induces non-vanishing third-order mixed derivatives, which in turn generate curvature in the dual connections. Numerical experiments on 2D Gaussian (ε_max = 9.75) and 4-category categorical (ε_max = 0.435) manifolds confirm the theoretical predictions with machine-precision agreement. These results provide a minimal sufficient condition for curvature emergence and offer practical guidance for the design of geometrically faithful divergence modifications.</p> |
| title | Beyond Dual Flatness: Curvature Emergence via Anisotropic Metric Perturbations |
| topic | information geometry dual flatness Pythagorean theorem metric perturbation dual connections curvature emergence Bregman divergence statistical manifolds |
| url | https://doi.org/10.5281/zenodo.18019235 |