{
  "id": "2203.09677",
  "version": "v1",
  "published": "2022-03-18T01:11:36.000Z",
  "updated": "2022-03-18T01:11:36.000Z",
  "title": "Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities",
  "authors": [
    "Artur O. Lopes",
    "Rafael O. Ruggiero"
  ],
  "comment": "Keywords: Geodesics; infinite-dimensional Riemannian manifold; equilibrium probabilities; KL-divergence; information projections; Pythagorean inequalities; Fourier-like basis",
  "categories": [
    "math.DS",
    "cs.IT",
    "math-ph",
    "math.DG",
    "math.IT",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider here the discrete time dynamics described by a transformation $T:M \\to M$, where $T$ is either the action of shift $T=\\sigma$ 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 $C^{1+\\alpha}$ (\\,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 H\\\"older 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 $T=\\sigma$ and $M=\\{0,1\\}^\\mathbb{N}$ such basis exists. In a different direction, we also consider the KL-divergence $D_{KL}(\\mu_1,\\mu_2)$ for a pair of equilibrium probabilities. If $D_{KL}(\\mu_1,\\mu_2)=0$, then $\\mu_1=\\mu_2$. Although $D_{KL}$ is not a metric in $\\mathcal{N}$, it describes the proximity between $\\mu_1$ and $\\mu_2$. A natural problem is: for a fixed probability $\\mu_1\\in \\mathcal{N}$ consider the probability $\\mu_2$ in a convex set of probabilities in $\\mathcal{N}$ which minimizes $D_{KL}(\\mu_1,\\mu_2)$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-18T01:11:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37A60",
      "94A15",
      "94A17"
    ],
    "keywords": [
      "probability",
      "hölder equilibrium probabilities",
      "dynamical information projections",
      "discrete time dynamics",
      "natural riemannian metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}