概率几何的凸坐标图表族逼近
Approximation of Probability Geometries via Convex Families of Coordinate Charts
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中文总结 AI 辅助
本文针对概率坐标几何的无限维可容许图类，提出基于斜坡函数凸组合的有限维逼近理论，并给出斜坡插值与伯恩斯坦逼近两种方案及重心误差的稳定性界。
中文摘要 AI 辅助
在概率坐标几何中，限制在区间上的连续分布函数可作为概率坐标图。在这些坐标中取平均定义了概率重心及相关的科尔莫戈罗夫矩。由于可容许图类为无限维，本文发展了有限维逼近理论。相对于固定基图，同一区间上的每个图可由单位区间的严格递增变形唯一表示。证明了单调分段线性变形是基本斜坡函数的精确凸组合，从而通过单纯形中的分配向量参数化概率几何。此类几何在可容许类中是稠密的。我们研究了两种构造方案：斜坡插值和伯恩斯坦逼近。伯恩斯坦逼近也允许斜坡字典表示，其光滑贝塔原子由均匀次序统计量导出，同时自动保持可容许性。变形映射的一致逼近为概率重心和初始科尔莫戈罗夫矩提供了显式稳定性界。对于变形固有图，重心是分位数，其水平仅依赖于变形，将构造与基于图的量化估计联系起来。确定性数值示例展示了两种逼近方案及其诱导的重心误差。
英文摘要
Within probability-coordinate geometry, a continuous distribution function restricted to an interval can serve as a probability coordinate chart. Averaging in these coordinates defines probability barycenters and associated Kolmogorov moments. Because the class of admissible charts is infinite dimensional, this paper develops a finite-dimensional approximation theory. Relative to a fixed base chart, every chart on the same interval is represented uniquely by a strictly increasing deformation of the unit interval. Monotone piecewise-linear deformations are shown to be exact convex combinations of elementary ramp functions, thereby parametrising probability geometries by allocation vectors in a simplex. Such geometries are dense in the admissible class. We study two constructive schemes: ramp interpolation and Bernstein approximation. Bernstein approximants also admit a ramp-dictionary representation with smooth beta atoms derived from uniform order statistics, while automatically preserving admissibility. Uniform approximation of deformation maps yields explicit stability bounds for probability barycenters and initial Kolmogorov moments. For deformed intrinsic charts, the barycenter is a quantile whose level depends only on the deformation, connecting the construction with chart-based quantile estimation. Deterministic numerical illustrations demonstrate both approximation schemes and their induced barycenter errors.
