Sharp Relaxed Triangle Inequality for Rényi Divergence of Order 1/2 Between Univariate Gaussians
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| Format: | Recurso digital |
| Language: | English |
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2026
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| _version_ | 1866901691128872960 |
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| author | Alex Shvets |
| author_facet | Alex Shvets |
| contents | <p>For Kullback–Leibler divergence between multivariate Gaussians, a sharp relaxed triangle inequality was recently obtained by Xiao et al. (arXiv:2602.02577v1, 2026). This paper develops an analogous extremal program for the Rényi divergence of order α = 1/2 in the one-dimensional Gaussian family. We derive a complete budget decomposition, prove an explicit closed-form optimum under a mirror-symmetry ansatz via a transcendental equation (S = 2 sinh(2s*)), establish that the supremum cannot be achieved at zero means, eliminate the plus-branch sector, and reduce the full mirror-symmetry conjecture to a single explicit Schur-concavity inequality. Extensions to general Rényi order α and to higher dimensions under commuting covariances are also provided.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18532134 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Sharp Relaxed Triangle Inequality for Rényi Divergence of Order 1/2 Between Univariate Gaussians Alex Shvets Rényi divergence relaxed triangle inequality Gaussian distributions information theory Schur-concavity Bhattacharyya coefficient quasi-metric <p>For Kullback–Leibler divergence between multivariate Gaussians, a sharp relaxed triangle inequality was recently obtained by Xiao et al. (arXiv:2602.02577v1, 2026). This paper develops an analogous extremal program for the Rényi divergence of order α = 1/2 in the one-dimensional Gaussian family. We derive a complete budget decomposition, prove an explicit closed-form optimum under a mirror-symmetry ansatz via a transcendental equation (S = 2 sinh(2s*)), establish that the supremum cannot be achieved at zero means, eliminate the plus-branch sector, and reduce the full mirror-symmetry conjecture to a single explicit Schur-concavity inequality. Extensions to general Rényi order α and to higher dimensions under commuting covariances are also provided.</p> |
| title | Sharp Relaxed Triangle Inequality for Rényi Divergence of Order 1/2 Between Univariate Gaussians |
| topic | Rényi divergence relaxed triangle inequality Gaussian distributions information theory Schur-concavity Bhattacharyya coefficient quasi-metric |
| url | https://doi.org/10.5281/zenodo.18532134 |