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Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Artur O. Lopes, Rafael O. Ruggiero

Abstract

We consider here the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is either the action of shift $T=σ$ on the symbolic space $M=\{1,2,...,d\}^\mathbb{N}$, or, $T$ describes the action of a $d$ to $1$ expanding transformation $T:S^1 \to S^1$ of class $C^{1+α}$ (\,for example $x \to T(x) =d\, x $ (mod $1) $\,), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibrium probabilities for Hölder potentials $A:M \to \mathbb{R}$ is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When $T=σ$ and $M=\{0,1\}^\mathbb{N}$ such basis exists. In a different direction, we also consider the KL-divergence $D_{KL}(μ_1,μ_2)$ for a pair of equilibrium probabilities. If $D_{KL}(μ_1,μ_2)=0$, then $μ_1=μ_2$. Although $D_{KL}$ is not a metric in $\mathcal{N}$, it describes the proximity between $μ_1$ and $μ_2$. A natural problem is: for a fixed probability $μ_1\in \mathcal{N}$ consider the probability $μ_2$ in a convex set of probabilities in $\mathcal{N}$ which minimizes $D_{KL}(μ_1,μ_2)$. This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in $\mathcal{N}$, a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.

Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Abstract

We consider here the discrete time dynamics described by a transformation , where is either the action of shift on the symbolic space , or, describes the action of a to expanding transformation of class (\,for example (mod \,), where is the unit circle. It is known that the infinite-dimensional manifold of equilibrium probabilities for Hölder potentials is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When and such basis exists. In a different direction, we also consider the KL-divergence for a pair of equilibrium probabilities. If , then . Although is not a metric in , it describes the proximity between and . A natural problem is: for a fixed probability consider the probability in a convex set of probabilities in which minimizes . This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in , a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.
Paper Structure (26 sections, 36 theorems, 227 equations, 2 figures)

This paper contains 26 sections, 36 theorems, 227 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries for the study of geodesics in $\mathcal{N}$
  3. Basics of Riemannian Geometry
  4. Preliminaries of the analytic structure of the set of normalized potentials
  5. Fundamental formulae from Thermodynamic Formalism
  6. On the Calculus of Thermodynamical formalism
  7. The existence of geodesics in $\mathcal{N}$
  8. Good Coordinate systems for the manifold of normalized potentials
  9. A system of partial differential equations for geodesic vector fields
  10. Uniform bounds for the PDE geodesic systems in a neighborhood of a Fourier-like probability
  11. First order systems of ordinary differential equations equivalent to first order PDE's
  12. On the existence and uniqueness of solutions of differential equations in $\mathcal{N}$
  13. Geodesic accessibility of the set of potentials associated to Fourier-like equilibrium measures on symbolic spaces with two symbols
  14. On the surface of Markov probabilities depending on two parameters
  15. KL-divergence and dynamical information projections
  16. ...and 11 more sections

Key Result

Theorem 1.2

Given $M$, $T$ and a Hölder normalized potential $A \in \mathcal{N}$, suppose there exist a Fourier-like Hilbert basis for the kernel of the Ruelle operator $\mathscr{L}_A$. Then, there exists an open ball $B_{r}(A)$ around $A$ such that for every $Q \in B_{r}(A)$ and every unit vector $X \in T_{B}\

Figures (2)

  • Figure 1: Numerical simulation - geodesics emanating from the point $(1/2,1/2)$ on parameter coordinates $(r,s)\in (0,1)\times (0,1)$ which describes the set of Markov probabilities.
  • Figure 2: Numerical simulation - geodesics emanating from the point $(0.35,0.15)$ on parameter coordinates $(r,s)\in (0,1)\times (0,1)$ which describes the set of Markov probabilities.

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 3.1
  • ...and 51 more