13,722 research outputs found
Stein's method for the half-normal distribution with applications to limit theorems related to the simple symmetric random walk
We develop Stein's method for the half-normal distribution and apply it to
derive rates of convergence in distributional limit theorems for three
statistics of the simple symmetric random walk: the maximum value, the number
of returns to the origin and the number of sign changes up to a given time .
We obtain explicit error bounds with the optimal rate for both the
Kolmogorov and the Wasserstein metric. In order to apply Stein's method, we
compare the characterizing operator of the limiting half-normal distribution
with suitable characterizations of the discrete approximating distributions,
exploiting a recent technique by Goldstein and Reinert \cite{GolRei13}.Comment: 22 pages, final version, results unchange
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Local limit theorems via Landau-Kolmogorov inequalities
In this article, we prove new inequalities between some common probability
metrics. Using these inequalities, we obtain novel local limit theorems for the
magnetization in the Curie-Weiss model at high temperature, the number of
triangles and isolated vertices in Erd\H{o}s-R\'{e}nyi random graphs, as well
as the independence number in a geometric random graph. We also give upper
bounds on the rates of convergence for these local limit theorems and also for
some other probability metrics. Our proofs are based on the Landau-Kolmogorov
inequalities and new smoothing techniques.Comment: Published at http://dx.doi.org/10.3150/13-BEJ590 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Quantifying uncertainties on excursion sets under a Gaussian random field prior
We focus on the problem of estimating and quantifying uncertainties on the
excursion set of a function under a limited evaluation budget. We adopt a
Bayesian approach where the objective function is assumed to be a realization
of a Gaussian random field. In this setting, the posterior distribution on the
objective function gives rise to a posterior distribution on excursion sets.
Several approaches exist to summarize the distribution of such sets based on
random closed set theory. While the recently proposed Vorob'ev approach
exploits analytical formulae, further notions of variability require Monte
Carlo estimators relying on Gaussian random field conditional simulations. In
the present work we propose a method to choose Monte Carlo simulation points
and obtain quasi-realizations of the conditional field at fine designs through
affine predictors. The points are chosen optimally in the sense that they
minimize the posterior expected distance in measure between the excursion set
and its reconstruction. The proposed method reduces the computational costs due
to Monte Carlo simulations and enables the computation of quasi-realizations on
fine designs in large dimensions. We apply this reconstruction approach to
obtain realizations of an excursion set on a fine grid which allow us to give a
new measure of uncertainty based on the distance transform of the excursion
set. Finally we present a safety engineering test case where the simulation
method is employed to compute a Monte Carlo estimate of a contour line
Causal Inference by Stochastic Complexity
The algorithmic Markov condition states that the most likely causal direction
between two random variables X and Y can be identified as that direction with
the lowest Kolmogorov complexity. Due to the halting problem, however, this
notion is not computable.
We hence propose to do causal inference by stochastic complexity. That is, we
propose to approximate Kolmogorov complexity via the Minimum Description Length
(MDL) principle, using a score that is mini-max optimal with regard to the
model class under consideration. This means that even in an adversarial
setting, such as when the true distribution is not in this class, we still
obtain the optimal encoding for the data relative to the class.
We instantiate this framework, which we call CISC, for pairs of univariate
discrete variables, using the class of multinomial distributions. Experiments
show that CISC is highly accurate on synthetic, benchmark, as well as
real-world data, outperforming the state of the art by a margin, and scales
extremely well with regard to sample and domain sizes
Comparison of some Reduced Representation Approximations
In the field of numerical approximation, specialists considering highly
complex problems have recently proposed various ways to simplify their
underlying problems. In this field, depending on the problem they were tackling
and the community that are at work, different approaches have been developed
with some success and have even gained some maturity, the applications can now
be applied to information analysis or for numerical simulation of PDE's. At
this point, a crossed analysis and effort for understanding the similarities
and the differences between these approaches that found their starting points
in different backgrounds is of interest. It is the purpose of this paper to
contribute to this effort by comparing some constructive reduced
representations of complex functions. We present here in full details the
Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM)
together with other approaches that enter in the same category
A quantitative central limit theorem for the effective conductance on the discrete torus
We study a random conductance problem on a -dimensional discrete torus of
size . The conductances are independent, identically distributed random
variables uniformly bounded from above and below by positive constants. The
effective conductance of the network is a random variable, depending on
, and the main result is a quantitative central limit theorem for this
quantity as . In terms of scalings we prove that this nonlinear
nonlocal function essentially behaves as if it were a simple spatial
average of the conductances (up to logarithmic corrections). The main
achievement of this contribution is the precise asymptotic description of the
variance of .Comment: 37 page
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