Approximation of Probability Geometries via Convex Families of Coordinate Charts
Abstract
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.
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BibTeX
@misc{cojocea2026approximation,
title = {{Approximation of Probability Geometries via Convex Families of Coordinate Charts}},
author = {Manuela-Simona Cojocea},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00039},
eprint = {2610.00039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00039},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Approximation of Probability Geometries via Convex Families of Coordinate Charts
AU - Manuela-Simona Cojocea
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00039
UR - https://arxiv.org/abs/2610.00039
AB - 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.
ER -