250 research outputs found
Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation
We give the first rigorous proof of the convergence of Riemannian Hamiltonian
Monte Carlo, a general (and practical) method for sampling Gibbs distributions.
Our analysis shows that the rate of convergence is bounded in terms of natural
smoothness parameters of an associated Riemannian manifold. We then apply the
method with the manifold defined by the log barrier function to the problems of
(1) uniformly sampling a polytope and (2) computing its volume, the latter by
extending Gaussian cooling to the manifold setting. In both cases, the total
number of steps needed is O^{*}(mn^{\frac{2}{3}}), improving the state of the
art. A key ingredient of our analysis is a proof of an analog of the KLS
conjecture for Gibbs distributions over manifolds
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
The Discrete Fundamental Group of the Associahedron, and the Exchange Module
The associahedron is an object that has been well studied and has numerous
applications, particularly in the theory of operads, the study of non-crossing
partitions, lattice theory and more recently in the study of cluster algebras.
We approach the associahedron from the point of view of discrete homotopy
theory. We study the abelianization of the discrete fundamental group, and show
that it is free abelian of rank . We also find a combinatorial
description for a basis of this rank. We also introduce the exchange module of
the type cluster algebra, used to model the relations in the cluster
algebra. We use the discrete fundamental group to the study of exchange module,
and show that it is also free abelian of rank .Comment: 16 pages, 4 figure
Fast MCMC sampling algorithms on polytopes
We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and
the John walk, for generating samples from the uniform distribution over a
polytope. Both random walks are sampling algorithms derived from interior point
methods. The former is based on volumetric-logarithmic barrier introduced by
Vaidya whereas the latter uses John's ellipsoids. We show that the Vaidya walk
mixes in significantly fewer steps than the logarithmic-barrier based Dikin
walk studied in past work. For a polytope in defined by
linear constraints, we show that the mixing time from a warm start is bounded
as , compared to the mixing time
bound for the Dikin walk. The cost of each step of the Vaidya walk is of the
same order as the Dikin walk, and at most twice as large in terms of constant
pre-factors. For the John walk, we prove an
bound on its mixing time and conjecture
that an improved variant of it could achieve a mixing time of
. Additionally, we propose variants
of the Vaidya and John walks that mix in polynomial time from a deterministic
starting point. The speed-up of the Vaidya walk over the Dikin walk are
illustrated in numerical examples.Comment: 86 pages, 9 figures, First two authors contributed equall
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