A Unified Approach to Aitchison’s, Dually Affine, and Transport Geometries of the Probability Simplex
A critical processing step for AI algorithms is mapping the raw data to a landscape where the similarity of two data points is conveniently defined. Frequently, when the data points are compositions of probability functions, the similarity is reduced to affine geometric concepts; the basic notion is...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/12/823 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | A critical processing step for AI algorithms is mapping the raw data to a landscape where the similarity of two data points is conveniently defined. Frequently, when the data points are compositions of probability functions, the similarity is reduced to affine geometric concepts; the basic notion is that of the straight line connecting two data points, defined as a zero-acceleration line segment. This paper provides an axiomatic presentation of the probability simplex’s most commonly used affine geometries. One result is a coherent presentation of gradient flow in Aichinson’s compositional data, Amari’s information geometry, the Kantorivich distance, and the Lagrangian optimization of the probability simplex. |
|---|---|
| ISSN: | 2075-1680 |