Beyond Nonexpansive Operations in Quantitative Algebraic Reasoning

TL;DR

This paper extends quantitative equational logic to handle algebras with generalized metric structures and operations that are nonexpansive up to a lifting, with applications to probability distribution distances in machine learning.

Contribution

It introduces a generalized framework for quantitative algebraic reasoning, broadening the scope beyond nonexpansive operations, and applies it to the { extL}ukaszyk--Karmowski distance.

Findings

Extended the framework to generalized metric spaces

Applied the framework to probability distribution distances

Enhanced reasoning capabilities in quantitative algebra

Abstract

The framework of quantitative equational logic has been successfully applied to reason about algebras whose carriers are metric spaces and operations are nonexpansive. We extend this framework in two orthogonal directions: algebras endowed with generalised metric space structures, and operations being nonexpansive up to a lifting. We apply our results to the algebraic axiomatisation of the {\L}ukaszyk--Karmowski distance on probability distributions, which has recently found application in the field of representation learning on Markov processes.