494 research outputs found
Long and short paths in uniform random recursive dags
In a uniform random recursive k-dag, there is a root, 0, and each node in
turn, from 1 to n, chooses k uniform random parents from among the nodes of
smaller index. If S_n is the shortest path distance from node n to the root,
then we determine the constant \sigma such that S_n/log(n) tends to \sigma in
probability as n tends to infinity. We also show that max_{1 \le i \le n}
S_i/log(n) tends to \sigma in probability.Comment: 16 page
Integration of Langevin Equations with Multiplicative Noise and Viability of Field Theories for Absorbing Phase Transitions
Efficient and accurate integration of stochastic (partial) differential
equations with multiplicative noise can be obtained through a split-step
scheme, which separates the integration of the deterministic part from that of
the stochastic part, the latter being performed by sampling exactly the
solution of the associated Fokker-Planck equation. We demonstrate the
computational power of this method by applying it to most absorbing phase
transitions for which Langevin equations have been proposed. This provides
precise estimates of the associated scaling exponents, clarifying the
classification of these nonequilibrium problems, and confirms or refutes some
existing theories.Comment: 4 pages. 4 figures. RevTex. Slightly changed versio
Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem
We study a random bisection problem where an initial interval of length x is
cut into two random fragments at the first stage, then each of these two
fragments is cut further, etc. We compute the probability P_n(x) that at the
n-th stage, each of the 2^n fragments is shorter than 1. We show that P_n(x)
approaches a traveling wave form, and the front position x_n increases as
x_n\sim n^{\beta}{\rho}^n for large n. We compute exactly the exponents
\rho=1.261076... and \beta=0.453025.... as roots of transcendental equations.
We also solve the m-section problem where each interval is broken into m
fragments. In particular, the generalized exponents grow as \rho_m\approx
m/(\ln m) and \beta_m\approx 3/(2\ln m) in the large m limit. Our approach
establishes an intriguing connection between extreme value statistics and
traveling wave propagation in the context of the fragmentation problem.Comment: 4 pages Revte
Non-parametric comparison of histogrammed two-dimensional data distributions using the Energy Test
When monitoring complex experiments, comparison is often made between regularly acquired histograms of data and reference histograms which represent the ideal state of the equipment. With the larger HEP experiments now ramping up, there is a need for automation of this task since the volume of comparisons could overwhelm human operators. However, the two-dimensional histogram comparison tools available in ROOT have been noted in the past to exhibit shortcomings. We discuss a newer comparison test for two-dimensional histograms, based on the Energy Test of Aslan and Zech, which provides more conclusive
discrimination between histograms of data coming from different distributions than methods provided in a recent ROOT release.The Science and Technology Facilities Council, U
PAC-Bayesian Bounds for Randomized Empirical Risk Minimizers
The aim of this paper is to generalize the PAC-Bayesian theorems proved by
Catoni in the classification setting to more general problems of statistical
inference. We show how to control the deviations of the risk of randomized
estimators. A particular attention is paid to randomized estimators drawn in a
small neighborhood of classical estimators, whose study leads to control the
risk of the latter. These results allow to bound the risk of very general
estimation procedures, as well as to perform model selection
The sphere and the geometric substratum of power law probability distributions
Links between power law probability distributions and marginal distributions
of uniform laws on -spheres in show that a mathematical
derivation of the Boltzmann-Gibbs distribution necessarily passes through power
law ones. Results are also given that link parameters and to the value
of the non-extensivity parameter that characterizes these power laws in the
context of non-extensive statistics.Comment: 10 page
Concentration inequalities for random fields via coupling
We present a new and simple approach to concentration inequalities for
functions around their expectation with respect to non-product measures, i.e.,
for dependent random variables. Our method is based on coupling ideas and does
not use information inequalities. When one has a uniform control on the
coupling, this leads to exponential concentration inequalities. When such a
uniform control is no more possible, this leads to polynomial or
stretched-exponential concentration inequalities. Our abstract results apply to
Gibbs random fields, in particular to the low-temperature Ising model which is
a concrete example of non-uniformity of the coupling.Comment: New corrected version; 22 pages; 1 figure; New result added:
stretched-exponential inequalit
Stationary probability density of stochastic search processes in global optimization
A method for the construction of approximate analytical expressions for the
stationary marginal densities of general stochastic search processes is
proposed. By the marginal densities, regions of the search space that with high
probability contain the global optima can be readily defined. The density
estimation procedure involves a controlled number of linear operations, with a
computational cost per iteration that grows linearly with problem size
A novel approach to light-front perturbation theory
We suggest a possible algorithm to calculate one-loop n-point functions
within a variant of light-front perturbation theory. The key ingredients are
the covariant Passarino-Veltman scheme and a surprising integration formula
that localises Feynman integrals at vanishing longitudinal momentum. The
resulting expressions are generalisations of Weinberg's infinite-momentum
results and are manifestly Lorentz invariant. For n = 2 and 3 we explicitly
show how to relate those to light-front integrals with standard energy
denominators. All expressions are rendered finite by means of transverse
dimensional regularisation.Comment: 10 pages, 5 figure
Robustness and Generalization
We derive generalization bounds for learning algorithms based on their
robustness: the property that if a testing sample is "similar" to a training
sample, then the testing error is close to the training error. This provides a
novel approach, different from the complexity or stability arguments, to study
generalization of learning algorithms. We further show that a weak notion of
robustness is both sufficient and necessary for generalizability, which implies
that robustness is a fundamental property for learning algorithms to work
- …