Geometry of measures on smoothly stratified metric spaces

Jonathan C. Mattingly; Ezra Miller; Do Tran

Geometry of measures on smoothly stratified metric spaces

Metric Geometry 2023-11-17 v1 Probability Statistics Theory Statistics Theory

Authors: Jonathan C. Mattingly , Ezra Miller , Do Tran

Abstract

Any measure $\mu$ on a CAT(k) space M that is stratified as a finite union of manifolds and has local exponential maps near the Fr\'echet mean $\bar\mu$ yields a continuous "tangential collapse" from the tangent cone of M at $\bar\mu$ to a vector space that preserves the Fr\'echet mean, restricts to an isometry on the "fluctuating cone" of directions in which the Fr\'echet mean can vary under perturbation of $\mu$ , and preserves angles between arbitrary and fluctuating tangent vectors at the Fr\'echet mean.

Keywords

cat(k) spaces stratified metric spaces measure geometry fréchet mean tangential collapse exponential map

Cite

@article{arxiv.2311.09453,
  title  = {Geometry of measures on smoothly stratified metric spaces},
  author = {Jonathan C. Mattingly and Ezra Miller and Do Tran},
  journal= {arXiv preprint arXiv:2311.09453},
  year   = {2023}
}

Comments

32 pages, 4 figures. This is the second in a four-part series starting with shadow geometry and leading to central limit theorems on stratified spaces

Geometry of measures on smoothly stratified metric spaces

Abstract

Keywords

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