多项式与负多项式势的同伦几何
A Homotopical Geometry of the Multinomial and Negative-Multinomial Potentials
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中文总结 AI 辅助
本文研究多项式与负多项式势凸组合形成的 Hessian 度量族，证明其正定性、曲率符号变化、复合分布解释及广义毕达哥拉斯定理，揭示其几何与概率结构。
中文摘要 AI 辅助
我们研究由多项式与负多项式 Fisher 信息 Hessian 的 log-配分函数（而非密度）凸组合形成的一参数 Hessian 度量族 $g_\alpha=\alpha h_+ +(1-\alpha)h_-$，$\alpha\in[0,1]$。我们完整证明了四个结果。（1）对所有 $\alpha\in[0,1]$，$g_\alpha\succ0$；其高斯曲率（$n=2$）在 $\alpha=0$ 时为 $-\tfrac14$，在 $\alpha=1$ 时为 $+\tfrac14$，但对中间 $\alpha$ 值非恒定，并在闭式临界值 $\alpha^\ast(w)$ 处改变符号，且具有解析唯一性证明。（2）将 $g_\alpha$ 视为 Kasner 型体积方程的空间度量，正定性强制一个守恒量 $\beta\le0$，而第二个独立量 $C=4\det K$ 控制方程的分支（有界振荡、抛物型或无界增长），且不由 $\beta$ 或正定性单独决定；我们证明该判别式字典是关于抽象矩阵方程的定理，$g_\alpha$ 提供 $C$ 的任一种符号的初始数据，而轨迹不必停留在 $\{g_\alpha(\theta)\}$ 上。（3）$\Psi_\alpha$ 被识别为复合分布的 log-配分函数：一个负二项潜在计数 $M$（参数 $r=1-\alpha$），辅以 Bernoulli 奇偶校验位并按多项式分裂，且对每个 $\alpha$ 具有非负基测度。（4）关联的 Bregman 散度满足广义毕达哥拉斯定理，其到对称轨迹上的投影是算术平均，与 $\alpha$ 无关；端点的曲率符号与真实测地三角形的 Gauss-Bonnet 盈余相匹配。数值检验始终与分析证明分开进行。
英文摘要
We study the one-parameter family of Hessian metrics $g_α=αh_++(1-α)h_-$, $α\in[0,1]$, formed by convexly combining the log-partition functions (not the densities) of the multinomial and negative multinomial Fisher information Hessians. Four results are proved in full. (1) $g_α\succ0$ for all $α\in[0,1]$; its Gaussian curvature ($n=2$) is $-\tfrac14$ at $α=0$ and $+\tfrac14$ at $α=1$, but non-constant for intermediate $α$, changing sign at a closed-form critical value $α^\ast(w)$, with an analytic uniqueness proof. (2) Viewing $g_α$ as the spatial metric of a Kasner-type volume equation, positive-definiteness forces one conserved quantity $β\le0$, while a second, independent quantity $C=4\det K$ governs the equation's branch (bounded oscillatory, parabolic, or unbounded growth) and is not fixed by $β$ or positive-definiteness alone; we show this discriminant dictionary is a theorem about the abstract matrix equation, to which $g_α$ supplies initial data of either sign of $C$ without the trajectories remaining on $\{g_α(θ)\}$ itself. (3) $Ψ_α$ is identified as the log-partition function of a compound distribution: a negative-binomial latent count $M$ (parameter $r=1-α$), augmented by a Bernoulli parity bit and split multinomially, with non-negative base measure for every $α$. (4) The associated Bregman divergence obeys a generalised Pythagorean theorem whose projection onto the symmetric locus is the arithmetic mean, independent of $α$; the endpoints' curvature sign matches the Gauss--Bonnet excess of genuine geodesic triangles. Numerical checks are kept separate from analytic proofs throughout.
发表机构
- Nagoya Mathematical and Information Science Research（名古屋数学与信息科学研究）
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