Gibbs/Metropolis algorithms on a convex polytope

Persi Diaconis (1) , Gilles Lebeau (2) , Laurent Michel (2)

1 Department of Statistics [Stanford]
2 JAD - Laboratoire Jean Alexandre Dieudonné

This paper gives sharp rates of convergence for natural versions of the Metropolis algorithm for sampling from the uniform distribution on a convex polytope. The singular proposal distribution , based on a walk moving locally in one of a fixed, finite set of directions, needs some new tools. We get useful bounds on the spectrum and eigenfunctions using Nash and Weyl-type inequalities. The top eigenvalues of the Markov chain are closely related to the Neuman eigenvalues of the polytope for a novel Laplacian.

Domaines: Probabilités [math.PR]; Théorie spectrale [math.SP]

Persi Diaconis, Gilles Lebeau, Laurent Michel. Gibbs/Metropolis algorithms on a convex polytope. Mathematische Zeitschrift, 2012, 272 (1), pp.109-129. ⟨10.1007/s00209-011-0924-5⟩. ⟨hal-01284522⟩