11 research outputs found
On Sampling from the Gibbs Distribution with Random Maximum A-Posteriori Perturbations
In this paper we describe how MAP inference can be used to sample efficiently
from Gibbs distributions. Specifically, we provide means for drawing either
approximate or unbiased samples from Gibbs' distributions by introducing low
dimensional perturbations and solving the corresponding MAP assignments. Our
approach also leads to new ways to derive lower bounds on partition functions.
We demonstrate empirically that our method excels in the typical "high signal -
high coupling" regime. The setting results in ragged energy landscapes that are
challenging for alternative approaches to sampling and/or lower bounds
Exact thresholds for Ising-Gibbs samplers on general graphs
We establish tight results for rapid mixing of Gibbs samplers for the
Ferromagnetic Ising model on general graphs. We show that if
then there exists a constant C such that the discrete
time mixing time of Gibbs samplers for the ferromagnetic Ising model on any
graph of n vertices and maximal degree d, where all interactions are bounded by
, and arbitrary external fields are bounded by . Moreover, the
spectral gap is uniformly bounded away from 0 for all such graphs, as well as
for infinite graphs of maximal degree d. We further show that when
, with high probability over the Erdos-Renyi random graph
, it holds that the mixing time of Gibbs samplers is
Both results are tight, as it is known that
the mixing time for random regular and Erdos-Renyi random graphs is, with high
probability, exponential in n when , and ,
respectively. To our knowledge our results give the first tight sufficient
conditions for rapid mixing of spin systems on general graphs. Moreover, our
results are the first rigorous results establishing exact thresholds for
dynamics on random graphs in terms of spatial thresholds on trees.Comment: Published in at http://dx.doi.org/10.1214/11-AOP737 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Learning Loosely Connected Markov Random Fields
We consider the structure learning problem for graphical models that we call
loosely connected Markov random fields, in which the number of short paths
between any pair of nodes is small, and present a new conditional independence
test based algorithm for learning the underlying graph structure. The novel
maximization step in our algorithm ensures that the true edges are detected
correctly even when there are short cycles in the graph. The number of samples
required by our algorithm is C*log p, where p is the size of the graph and the
constant C depends on the parameters of the model. We show that several
previously studied models are examples of loosely connected Markov random
fields, and our algorithm achieves the same or lower computational complexity
than the previously designed algorithms for individual cases. We also get new
results for more general graphical models, in particular, our algorithm learns
general Ising models on the Erdos-Renyi random graph G(p, c/p) correctly with
running time O(np^5).Comment: 45 pages, minor revisio
The Ising Partition Function: Zeros and Deterministic Approximation
We study the problem of approximating the partition function of the
ferromagnetic Ising model in graphs and hypergraphs. Our first result is a
deterministic approximation scheme (an FPTAS) for the partition function in
bounded degree graphs that is valid over the entire range of parameters
(the interaction) and (the external field), except for the case
(the "zero-field" case). A randomized algorithm (FPRAS)
for all graphs, and all , has long been known. Unlike most other
deterministic approximation algorithms for problems in statistical physics and
counting, our algorithm does not rely on the "decay of correlations" property.
Rather, we exploit and extend machinery developed recently by Barvinok, and
Patel and Regts, based on the location of the complex zeros of the partition
function, which can be seen as an algorithmic realization of the classical
Lee-Yang approach to phase transitions. Our approach extends to the more
general setting of the Ising model on hypergraphs of bounded degree and edge
size, where no previous algorithms (even randomized) were known for a wide
range of parameters. In order to achieve this extension, we establish a tight
version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a
classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results
on optimality of univariate Lee-Yang theorem
Complexity of Ising Polynomials
This paper deals with the partition function of the Ising model from
statistical mechanics, which is used to study phase transitions in physical
systems. A special case of interest is that of the Ising model with constant
energies and external field. One may consider such an Ising system as a simple
graph together with vertex and edge weights. When these weights are considered
indeterminates, the partition function for the constant case is a trivariate
polynomial Z(G;x,y,z). This polynomial was studied with respect to its
approximability by L. A. Goldberg, M. Jerrum and M. Paterson in 2003.
Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied by D.
Andr\'{e}n and K. Markstr\"{o}m in 2009.
We consider the complexity of Z(G;t,y) and Z(G;x,y,z) in comparison to that
of the Tutte polynomial, which is well-known to be closely related to the Potts
model in the absence of an external field. We show that Z(G;\x,\y,\z) is
#P-hard to evaluate at all points in , except those in an
exception set of low dimension, even when restricted to simple graphs which are
bipartite and planar. A counting version of the Exponential Time Hypothesis,
#ETH, was introduced by H. Dell, T. Husfeldt and M. Wahl\'{e}n in 2010 in order
to study the complexity of the Tutte polynomial. In analogy to their results,
we give a dichotomy theorem stating that evaluations of Z(G;t,y) either take
exponential time in the number of vertices of to compute, or can be done in
polynomial time. Finally, we give an algorithm for computing Z(G;x,y,z) in
polynomial time on graphs of bounded clique-width, which is not known in the
case of the Tutte polynomial