137 research outputs found
Hitting Time of Quantum Walks with Perturbation
The hitting time is the required minimum time for a Markov chain-based walk
(classical or quantum) to reach a target state in the state space. We
investigate the effect of the perturbation on the hitting time of a quantum
walk. We obtain an upper bound for the perturbed quantum walk hitting time by
applying Szegedy's work and the perturbation bounds with Weyl's perturbation
theorem on classical matrix. Based on the definition of quantum hitting time
given in MNRS algorithm, we further compute the delayed perturbed hitting time
(DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the
upper bound for DPQHT is actually greater than the difference between the
square root of the upper bound for a perturbed random walk and the square root
of the lower bound for a random walk.Comment: 9 page
Approximation algorithms for the normalizing constant of Gibbs distributions
Consider a family of distributions where
means that . Here is the
proper normalizing constant, equal to . Then
is known as a Gibbs distribution, and is the
partition function. This work presents a new method for approximating the
partition function to a specified level of relative accuracy using only a
number of samples, that is, when
. This is a sharp improvement over previous, similar approaches that
used a much more complicated algorithm, requiring
samples.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1015 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Estimation in spin glasses: A first step
The Sherrington--Kirkpatrick model of spin glasses, the Hopfield model of
neural networks and the Ising spin glass are all models of binary data
belonging to the one-parameter exponential family with quadratic sufficient
statistic. Under bare minimal conditions, we establish the
-consistency of the maximum pseudolikelihood estimate of the natural
parameter in this family, even at critical temperatures. Since very little is
known about the low and critical temperature regimes of these extremely
difficult models, the proof requires several new ideas. The author's version of
Stein's method is a particularly useful tool. We aim to introduce these
techniques into the realm of mathematical statistics through an example and
present some open questions.Comment: Published in at http://dx.doi.org/10.1214/009053607000000109 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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