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Entropic curvature not comparable to other curvatures -- or is it?

Supanat Kamtue, Shiping Liu, Florentin Münch, Norbert Peyerimhoff

TL;DR

The paper investigates entropic curvature and its θ-variants for finite Markov chains, notably establishing explicit positive lower bounds for abelian Cayley graphs and universal lower bounds for general chains, along with perturbation results that compare entropic and Bakry-Émery curvatures. It introduces an adapted Γ-calculus based on the logarithmic mean and derives both infinite- and finite-dimensional curvature-dimension bounds, plus asymptotically sharp upper bounds for cycle graphs. The work demonstrates substantial noncomparability between entropic curvature and other notions such as Bakry-Émery, Ollivier Ricci, and sectional curvatures through concrete examples, and articulates perturbation tools and mapping-representation ideas to transfer curvature information across related chains. The contributions advance understanding of discrete curvature landscapes, provide explicit constants, and suggest directions for applying curvature concepts to MLSI, spectral gaps, and graph-structured data analysis.

Abstract

In this paper we consider global $θ$-curvatures of finite Markov chains with associated means $θ$ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-Émery curvature, we also allow for a finite dimension parameter by making use of an adapted $Γ$ calculus for $θ$-curvatures. We prove explicit positive lower curvature bounds (both finite- and infinite-dimensional) for finite abelian Cayley graphs. In the case of cycles, we provide also an upper curvature bound which shows that our lower bounds are asymptotically sharp (up to a logarithmic factor). Moreover, we prove new universal lower curvature bounds for finite Markov chains as well as curvature perturbation results (allowing, in particular, to compare entropic and Bakry-Émery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-Émery curvature or Ollivier Ricci and sectional curvatures.

Entropic curvature not comparable to other curvatures -- or is it?

TL;DR

The paper investigates entropic curvature and its θ-variants for finite Markov chains, notably establishing explicit positive lower bounds for abelian Cayley graphs and universal lower bounds for general chains, along with perturbation results that compare entropic and Bakry-Émery curvatures. It introduces an adapted Γ-calculus based on the logarithmic mean and derives both infinite- and finite-dimensional curvature-dimension bounds, plus asymptotically sharp upper bounds for cycle graphs. The work demonstrates substantial noncomparability between entropic curvature and other notions such as Bakry-Émery, Ollivier Ricci, and sectional curvatures through concrete examples, and articulates perturbation tools and mapping-representation ideas to transfer curvature information across related chains. The contributions advance understanding of discrete curvature landscapes, provide explicit constants, and suggest directions for applying curvature concepts to MLSI, spectral gaps, and graph-structured data analysis.

Abstract

In this paper we consider global -curvatures of finite Markov chains with associated means in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-Émery curvature, we also allow for a finite dimension parameter by making use of an adapted calculus for -curvatures. We prove explicit positive lower curvature bounds (both finite- and infinite-dimensional) for finite abelian Cayley graphs. In the case of cycles, we provide also an upper curvature bound which shows that our lower bounds are asymptotically sharp (up to a logarithmic factor). Moreover, we prove new universal lower curvature bounds for finite Markov chains as well as curvature perturbation results (allowing, in particular, to compare entropic and Bakry-Émery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-Émery curvature or Ollivier Ricci and sectional curvatures.
Paper Structure (24 sections, 21 theorems, 225 equations, 3 figures, 1 table)

This paper contains 24 sections, 21 theorems, 225 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our results
  3. Previous work on discrete curvature
  4. Setup and Notation
  5. Markov chains and means
  6. Relevant properties of the logarithmic mean
  7. Entropic curvature
  8. Adapted $\Gamma$ calculus and $\theta$-curvatures
  9. Positive curvature bounds for abelian Cayley graphs
  10. Special mapping representations and abelian Cayley graphs
  11. (Asymptotically) sharp lower curvature bound for infinite dimension
  12. Lower curvature bound for finite dimension
  13. (Asymptotically) sharp upper curvature bound for cycles
  14. Abelian Cayley graphs and Ricci flatness
  15. Universal lower curvature bounds and perturbation
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(X,Q,\pi)$ be an irreducible, reversible finite Markov chain such that the induced structure on $X$ is an abelian Cayley graph ${\rm{Cay}}(\Gamma,S)$ with finitely presented group $\Gamma$ and finite symmetric generating set $S$, and with the additional property Let $Q_{\min} := \min_{s \in S} Q(x,sx)$. Let $r:=\max_{s \in S} {\rm{ord}}(s)$ be the maximal order of the generators, and let $S_

Figures (3)

  • Figure 1: Contour plot and plot of the function $\log(g(s,t))$ over the domain $[-8\log(2),8\log(2)]\times[-8\log(2),8\log(2)]$, respectively, on the left. The values of the contours increase the closer they are to the point $(s,t)=(8\log(2),8\log(2))$. The graph of $g(s,s)$ over $[-8\log(2),8\log(2)]$ is plotted on the right.
  • Figure 2: Plot and contour plot of $b_0(e^t,1,e^s)\theta(e^t,1)+b_0(e^s,1,e^t)\theta(e^s,1)$ on the domains $[-2,1.1]\times[-2,1.1]$ and $[-3,1]\times[-3,1]$, respectively, on the left, and graph of the restriction of this function to the diagonal $s=t$ on $[-4,0]$ on the right.
  • Figure 3: A Ricci flat graph which is not (S)-Ricci flat, and therefore not an abelian Cayley graph (vertices labelled "F" and "R" are only Ricci-flat and only (R)-Ricci flat, respectively). The picture was generated via the Graph Curvature Calculator CKLLS-22.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 39 more