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A Fuzzy Decomposition Method to Establish Functional Inequalities

Suei-Wen Chen

Abstract

This paper presents a method to establish functional inequalities via fuzzy decomposition on the state space, which generalizes earlier results dealing with exact partitions of the state space. Given a reversible Markov chain on a finite state space, we define its projection chain and restriction chains from classes of a fuzzy partition on the state space. The Poincaré, log-Sobolev and modified log-Sobolev inequalities associated with the original chain can be estimated from those of its projection chain and restriction chains which tend to have simpler structures and hence easier to work with. An application of this generalization is presented in which the fuzzy decomposition method applies but its exact partitioning counterpart does not.

A Fuzzy Decomposition Method to Establish Functional Inequalities

Abstract

This paper presents a method to establish functional inequalities via fuzzy decomposition on the state space, which generalizes earlier results dealing with exact partitions of the state space. Given a reversible Markov chain on a finite state space, we define its projection chain and restriction chains from classes of a fuzzy partition on the state space. The Poincaré, log-Sobolev and modified log-Sobolev inequalities associated with the original chain can be estimated from those of its projection chain and restriction chains which tend to have simpler structures and hence easier to work with. An application of this generalization is presented in which the fuzzy decomposition method applies but its exact partitioning counterpart does not.

Paper Structure

This paper contains 6 sections, 1 theorem, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reversible Markov Chains and Functional Inequalities
  3. Decomposition Methods and Fuzzy Decomposition
  4. Main Result
  5. Related Works and Examples
  6. Proof

Key Result

Theorem 1.1

Let $(\Omega, \mathcal{P}(\Omega), \pi)$ be a finite probability space. Suppose we are given a fuzzy partition on $\Omega$ with classes $I$ which admits couplings $\kappa_{ij}$ between $\pi_i$ and $\pi_j$ for all $(i,j)\in I^2$ such that $\hat{Q}(i,j)>0$, with quality $\chi$. Then

Figures (1)

  • Figure 1: Illustration of the construction of $\mathcal{G}$

Theorems & Definitions (2)

  • Theorem 1.1
  • Example 2.1