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The max-type quasimetrics on probability simplices

Michał Eckstein, Tomasz Miller, Karol Życzkowski

TL;DR

The paper defines max-type quasimetrics on the probability simplex $\Delta_N$ via $D_f(P,Q)=\max_i (f(q_i)-f(p_i))$, with $f$ a strictly increasing function on $[0,1]$. It establishes that these quasimetrics induce the Euclidean topology and admit a Finslerian infinitesimal structure, making $(\Delta_N, D_f)$ a geodesic space; under suitable smoothness of $f$ the geodesics are as smooth as $f$. It further proves that $D_f$ is monotone under bistochastic maps, using a Finsler framework and the Birkhoff–von Neumann decomposition, thus extending Chebyshev-type distances to an asymmetric setting on probability distributions. These results suggest that max-type quasimetrics are a robust tool for directional distance problems on probability simplices, with potential applications in areas that rely on coordinatewise Chebyshev measures but require asymmetry.

Abstract

Quasimetric spaces form a natural framework to study distance problems with an inherent directional asymmetry. We introduce a simple novel class of quasimetrics on probability simplices, inspired by the Chebyshev distance. It is shown that such quasimetrics have expedient geometric properties -- they induce the Euclidean topology and a Finslerian infinitesimal structure, with which the probability simplices become geodesic spaces. Moreover, we prove that the broad family of the proposed quasimetrics are monotone under bistochastic maps.

The max-type quasimetrics on probability simplices

TL;DR

The paper defines max-type quasimetrics on the probability simplex via , with a strictly increasing function on . It establishes that these quasimetrics induce the Euclidean topology and admit a Finslerian infinitesimal structure, making a geodesic space; under suitable smoothness of the geodesics are as smooth as . It further proves that is monotone under bistochastic maps, using a Finsler framework and the Birkhoff–von Neumann decomposition, thus extending Chebyshev-type distances to an asymmetric setting on probability distributions. These results suggest that max-type quasimetrics are a robust tool for directional distance problems on probability simplices, with potential applications in areas that rely on coordinatewise Chebyshev measures but require asymmetry.

Abstract

Quasimetric spaces form a natural framework to study distance problems with an inherent directional asymmetry. We introduce a simple novel class of quasimetrics on probability simplices, inspired by the Chebyshev distance. It is shown that such quasimetrics have expedient geometric properties -- they induce the Euclidean topology and a Finslerian infinitesimal structure, with which the probability simplices become geodesic spaces. Moreover, we prove that the broad family of the proposed quasimetrics are monotone under bistochastic maps.

Paper Structure

This paper contains 4 sections, 13 theorems, 45 equations, 2 figures.

Table of Contents

  1. Quasimetric spaces
  2. Basic properties
  3. Finslerian structure
  4. Monotonicity

Key Result

Proposition 2.2

$(\Delta_N, D_f)$ defined above is a unitopological quasimetric space. Its topology is the standard Euclidean topology.

Figures (2)

  • Figure 1: The boundaries of certain forward (blue) and backward (red) quasimetric balls centered at $P = (2/9,1/3,4/9) \in \Delta_2$. Here $f(x) \vcentcolon = x^{1/3}$, what explains why $P$ seems off-center and different apparent sizes of the balls. For the definition of $C^\pm_i$ see Remark \ref{['rem_balls']}.
  • Figure 2: The same (boundaries of) quasimetric balls as in Figure \ref{['fig2']}, only seen from the side, with the points $P^\pm \in {\mathbb R}^3 \setminus \Delta_2$ visible.

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • ...and 26 more