信息几何中逆问题的两点方法
A two-point approach to the inverse problem in information geometry
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中文总结 AI 辅助
该研究在两点张量框架下求解一般度量仿射流形的信息几何逆问题，通过同伦约化得到相关对比函数，并将理论应用于特定流形的分析。
中文摘要 AI 辅助
我们在两点张量框架内构建信息几何中的逆问题，并针对一般度量仿射流形求解该问题，未施加任何曲率或挠率约束。该构造是显式的，直接从给定几何数据出发：度量张量与仿射结构配对，通过平行运输得到的局部平行性实现。这产生了诱导原始度量仿射流形的对比双形式。随后通过同伦约化，得到容许挠率的统计流形和统计流形的逆问题。由此获得合适的预对比函数和对比函数，包括统计流形与SMAT的若干已建立构造。我们将该一般理论应用于带有不变仿射联络的约化齐次伪黎曼流形，特别关注带有Cartan-Schouten联络的半单李群和带有Berger度量的奇维球面。
英文摘要
We formulate the inverse problem in information geometry within a two-point tensorial framework and solve it for general metric-affine manifolds, without imposing any curvature or torsion constraints. The construction is explicit and starts directly from the given geometric data: the metric tensor is paired to the affine structure, realized through a local parallelism obtained from parallel transport. This yields a contrast bi-form inducing the original metric-affine manifold. The inverse problems for statistical manifolds admitting torsion and for statistical manifolds are then recovered by homotopical reduction. In this way, suitable pre-contrast and contrast functions are obtained, including several established constructions for statistical manifolds and SMATs. We apply the general theory to reductive homogeneous pseudo-Riemannian manifolds endowed with invariant affine connections. Particular attention is devoted to semisimple Lie groups with Cartan-Schouten connections and to odd-dimensional spheres equipped with Berger metrics.
