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The Dual Burnside Process

Ivan Z. Feng

TL;DR

The paper introduces the Dual Burnside Process, a symmetry-aware Markov chain on $G^{*}$ with kernel $Q=AB$, paired with the classical Burnside chain on $X$ with kernel $K=BA$. It establishes a primal–dual spectral correspondence: the nonzero spectra of $Q$ and $K$ coincide, and mixing times differ by at most one step, with universal Doeblin floors that hold model-free. A comprehensive lumping framework is developed, enabling orbit- and conjugacy-class lumpings and transfer principles between $Q$ and $K$, including auxiliary-variable schemes and TV-preserving conditions. The authors instantiate the theory in two concrete symmetry models—value-permutation ($S_k$ on $[k]^n$) and coordinate-permutation ($S_n$ on $[k]^n$)—deriving explicit closed forms for $Q(g,h)$, stationary distributions, and lumped kernels, and establishing $n$-independent mixing bounds in key regimes. Collectively, the dual chain provides a conceptual mirror to the Burnside process and yields practical tools for symmetry-aware MCMC, with transferable insights to a broad class of group actions and lumping schemes.

Abstract

The Burnside process is a classical Markov chain for sampling uniformly from group orbits. We introduce the dual Burnside process, obtained by interchanging the roles of group elements and states. This dual chain has stationary law $π(g)\propto |X_g|$, is reversible, and admits a matrix factorization $Q=AB$, $K=BA$ with the classical Burnside kernel $K$. As a consequence the two chains share all nonzero eigenvalues and have mixing times that differ by at most one step. We further establish universal Doeblin floors, orbit and conjugacy-class lumpings, and transfer principles between $Q$ and $K$. We analyze explicit examples: the value-permutation model $S_k$ acting on $[k]^n$ and the coordinate-permutation model $S_n$ acting on $[k]^n$. These results show that the dual chain provides both a conceptual mirror to the classical Burnside process and practical advantages for symmetry-aware Markov chain Monte Carlo.

The Dual Burnside Process

TL;DR

The paper introduces the Dual Burnside Process, a symmetry-aware Markov chain on with kernel , paired with the classical Burnside chain on with kernel . It establishes a primal–dual spectral correspondence: the nonzero spectra of and coincide, and mixing times differ by at most one step, with universal Doeblin floors that hold model-free. A comprehensive lumping framework is developed, enabling orbit- and conjugacy-class lumpings and transfer principles between and , including auxiliary-variable schemes and TV-preserving conditions. The authors instantiate the theory in two concrete symmetry models—value-permutation ( on ) and coordinate-permutation ( on )—deriving explicit closed forms for , stationary distributions, and lumped kernels, and establishing -independent mixing bounds in key regimes. Collectively, the dual chain provides a conceptual mirror to the Burnside process and yields practical tools for symmetry-aware MCMC, with transferable insights to a broad class of group actions and lumping schemes.

Abstract

The Burnside process is a classical Markov chain for sampling uniformly from group orbits. We introduce the dual Burnside process, obtained by interchanging the roles of group elements and states. This dual chain has stationary law , is reversible, and admits a matrix factorization , with the classical Burnside kernel . As a consequence the two chains share all nonzero eigenvalues and have mixing times that differ by at most one step. We further establish universal Doeblin floors, orbit and conjugacy-class lumpings, and transfer principles between and . We analyze explicit examples: the value-permutation model acting on and the coordinate-permutation model acting on . These results show that the dual chain provides both a conceptual mirror to the classical Burnside process and practical advantages for symmetry-aware Markov chain Monte Carlo.

Paper Structure

This paper contains 37 sections, 65 theorems, 394 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Sampling Up to Symmetry
  3. The Dual Perspective
  4. Contributions and Organization
  5. Background and Preliminaries
  6. Notation and Conventions
  7. Group actions.
  8. Probability and Markov kernels.
  9. Re-indexing principle.
  10. Minorization.
  11. Lumping and lumpability.
  12. The Classical Burnside Process
  13. Literature Overview
  14. Main Theoretical Framework
  15. The Dual Burnside Process
  16. ...and 22 more sections

Key Result

Proposition 2.1

Let $\tilde{P}=\Pi^{1/2}P\Pi^{-1/2}=\tilde{P}^\top$ have eigenpairs $(\lambda_k,u_k)$ with $u_k^\top u_k=1$ and $1=\lambda_0\ge\lambda_1\ge\cdots\ge\lambda_{n-1}$. Then, for $k=0,\dots,n-1$,

Figures (2)

  • Figure 1: Primal–dual alternation via the legs $A$ and $B$.
  • Figure 2: Orbits of $\langle h\rangle$ and color counts $M_a(\phi)$.

Theorems & Definitions (161)

  • Proposition 2.1: Rayleigh–Ritz for the symmetric similarity
  • Corollary 2.2: Rayleigh–Ritz for $P$ in $L^2(\pi)$
  • proof
  • Remark 2.3
  • Proposition 2.4: Rosenthal’s bound; Rosenthal95
  • Remark 2.5
  • Definition 3.1: Dual Burnside process
  • Remark 3.2
  • Theorem 3.3: Universal dual stationary law
  • proof
  • ...and 151 more