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

Massively Parallel Construction of Radix Tree Forests for the Efficient Sampling of Discrete Probability Distributions

We compare different methods for sampling from discrete probability distributions and introduce a new algorithm which is especially efficient on massively parallel processors, such as GPUs. The scheme preserves the distribution properties of the input sequence, exposes constant time complexity on the average, and significantly lowers the average number of operations for certain distributions when sampling is performed in a parallel algorithm that requires synchronization afterwards. Avoiding load balancing issues of na\"ive approaches, a very efficient massively parallel construction algorithm for the required auxiliary data structure is complemented

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

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

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 Border Identification algorithm based on an “Anti-Bayesian” paradigm

Border Identification (BI) algorithms, a subset of Prototype Reduction Schemes (PRS) aim to reduce the number of training vectors so that the reduced set (the border set) contains only those patterns which lie near the border of the classes, and have sufficient information to perform a meaningful classification. However, one can see that the true border patterns (“near” border) are not able to perform the task independently as they are not able to always distinguish the testing samples. Thus, researchers have worked on this issue so as to find a way to strengthen the “border” set. A recent development in this field tries to add more border patterns, i.e., the “far” borders, to the border set, and this process continues until it reaches a stage at which the classification accuracy no longer increases. In this case, the cardinality of the border set is relatively high. In this paper, we aim to design a novel BI algorithm based on a new definition for the term “border”. We opt to select the patterns which lie at the border of the alternate class as the border patterns. Thus, those patterns which are neither on the true discriminant nor too close to the central position of the distributions, are added to the “border” set. The border patterns, which are very small in number (for example, five from both classes), selected in this manner, have the potential to perform a classification which is comparable to that obtained by well-known traditional classifiers like the SVM, and very close to the optimal Bayes’ bound

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

Simulation of truncated normal variables

We provide in this paper simulation algorithms for one-sided and two-sided truncated normal distributions. These algorithms are then used to simulate multivariate normal variables with restricted parameter space for any covariance structure.Comment: This 1992 paper appeared in 1995 in Statistics and Computing and the gist of it is contained in Monte Carlo Statistical Methods (2004), but I receive weekly requests for reprints so here it is

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

The p−p-sphere and the geometric substratum of power law probability distributions

Links between power law probability distributions and marginal distributions of uniform laws on $p$-spheres in $\mathbb{R}^{n}$ show that a mathematical derivation of the Boltzmann-Gibbs distribution necessarily passes through power law ones. Results are also given that link parameters $p$ and $n$ to the value of the non-extensivity parameter $q$ that characterizes these power laws in the context of non-extensive statistics.Comment: 10 page