Work

A Unified Approach to Aitchison’s, Dually Affine, and Transport Geometries of the Probability Simplex

Year: 2024

Type: article

Abstract: 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.

Source: Axioms

DOI: 10.3390/axioms13120823

Authors: Giovanni Pistone, Muhammad Shoaib

Institutions: Collegio Carlo Alberto, University of Genoa

Language: English

Topic: Mathematical Inequalities and Applications

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,728

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", European Commission, Collegio Carlo Alberto