135 research outputs found
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
Matrix norms and rapid mixing for spin systems
We give a systematic development of the application of matrix norms to rapid
mixing in spin systems. We show that rapid mixing of both random update Glauber
dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the
associated dependency matrix is less than 1. We give improved analysis for the
case in which the diagonal of the dependency matrix is (as in heat
bath dynamics). We apply the matrix norm methods to random update and
systematic scan Glauber dynamics for coloring various classes of graphs. We
give a general method for estimating a norm of a symmetric nonregular matrix.
This leads to improved mixing times for any class of graphs which is hereditary
and sufficiently sparse including several classes of degree-bounded graphs such
as nonregular graphs, trees, planar graphs and graphs with given tree-width and
genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Rapid mixing of Swendsen-Wang dynamics in two dimensions
We prove comparison results for the Swendsen-Wang (SW) dynamics, the
heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics
for the random-cluster model on arbitrary graphs. In particular, we prove that
rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum
degree and that rapid mixing of SW and rapid mixing of SB are equivalent.
Additionally, the spectral gap of SW and SB on planar graphs is bounded from
above and from below by the spectral gap of these dynamics on the corresponding
dual graph with suitably changed temperature.
As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the
Potts model on the two-dimensional square lattice at all non-critical
temperatures as well as rapid mixing for the two-dimensional Ising model at all
temperatures. Furthermore, we obtain new results for general graphs at high or
low enough temperatures.Comment: Ph.D. thesis, 66 page
Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees
We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter ? > 0; the special case ? = 1 corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete ?-regular tree for all ?. However, Restrepo et al. (2014) showed that for sufficiently large ? there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width.
We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of O(n) for the Glauber dynamics for unweighted independent sets on arbitrary trees. Moreover, for ? ? .44 we prove an optimal mixing time bound of O(n log n). We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree ?. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order ? = O(1/?). Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance/entropy via a non-trivial inductive proof
A graph polynomial for independent sets of bipartite graphs
We introduce a new graph polynomial that encodes interesting properties of
graphs, for example, the number of matchings and the number of perfect
matchings. Most importantly, for bipartite graphs the polynomial encodes the
number of independent sets (#BIS).
We analyze the complexity of exact evaluation of the polynomial at rational
points and show that for most points exact evaluation is #P-hard (assuming the
generalized Riemann hypothesis) and for the rest of the points exact evaluation
is trivial.
We conjecture that a natural Markov chain can be used to approximately
evaluate the polynomial for a range of parameters. The conjecture, if true,
would imply an approximate counting algorithm for #BIS, a problem shown, by
[Dyer et al. 2004], to be complete (with respect to, so called, AP-reductions)
for a rich logically defined sub-class of #P. We give a mild support for our
conjecture by proving that the Markov chain is rapidly mixing on trees. As a
by-product we show that the "single bond flip" Markov chain for the random
cluster model is rapidly mixing on constant tree-width graphs
Random-Cluster Dynamics in
The random-cluster model has been widely studied as a unifying framework for
random graphs, spin systems and electrical networks, but its dynamics have so
far largely resisted analysis. In this paper we analyze the Glauber dynamics of
the random-cluster model in the canonical case where the underlying graph is an
box in the Cartesian lattice . Our main result is a
upper bound for the mixing time at all values of the model
parameter except the critical point , and for all values of the
second model parameter . We also provide a matching lower bound proving
that our result is tight. Our analysis takes as its starting point the recent
breakthrough by Beffara and Duminil-Copin on the location of the random-cluster
phase transition in . It is reminiscent of similar results for
spin systems such as the Ising and Potts models, but requires the reworking of
several standard tools in the context of the random-cluster model, which is not
a spin system in the usual sense
A Unified Approach to Learning Ising Models: Beyond Independence and Bounded Width
We revisit the problem of efficiently learning the underlying parameters of
Ising models from data. Current algorithmic approaches achieve essentially
optimal sample complexity when given i.i.d. samples from the stationary measure
and the underlying model satisfies "width" bounds on the total
interaction involving each node. We show that a simple existing approach based
on node-wise logistic regression provably succeeds at recovering the underlying
model in several new settings where these assumptions are violated:
(1) Given dynamically generated data from a wide variety of local Markov
chains, like block or round-robin dynamics, logistic regression recovers the
parameters with optimal sample complexity up to factors. This
generalizes the specialized algorithm of Bresler, Gamarnik, and Shah [IEEE
Trans. Inf. Theory'18] for structure recovery in bounded degree graphs from
Glauber dynamics.
(2) For the Sherrington-Kirkpatrick model of spin glasses, given
independent samples, logistic regression recovers the
parameters in most of the known high-temperature regime via a simple reduction
to weaker structural properties of the measure. This improves on recent work of
Anari, Jain, Koehler, Pham, and Vuong [ArXiv'23] which gives distribution
learning at higher temperature.
(3) As a simple byproduct of our techniques, logistic regression achieves an
exponential improvement in learning from samples in the M-regime of data
considered by Dutt, Lokhov, Vuffray, and Misra [ICML'21] as well as novel
guarantees for learning from the adversarial Glauber dynamics of Chin, Moitra,
Mossel, and Sandon [ArXiv'23].
Our approach thus significantly generalizes the elegant analysis of Wu,
Sanghavi, and Dimakis [Neurips'19] without any algorithmic modification.Comment: 51 page
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