High-dimensional Probability Research Articles

Geometry of sets of quantum maps: A generic positive map acting on a high-dimensional system is not completely positive

We investigate the set (a) of positive, trace preserving maps acting on density matrices of size N and a sequence of its nested subsets: the sets of maps which are (b) decomposable, (c) completely positive, and (d) extended by identity impose positive partial transpose and (e) are superpositive. Working with the Hilbert–Schmidt (Euclidean) measure, we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiołkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity, high-dimensional probability, and geometry of Banach spaces, some of which are not standard.

Estimating multidimensional density functions for random variables in Wiener space

The Malliavin–Thalmaier (Springer Finance, Springer-Verlag, Berlin, 2006) formula was introduced for the simulation of high dimensional probability density functions. However, when this integration by parts formula is applied directly in computer simulations, we show that it is unstable. We propose an approximation to the Malliavin–Thalmaier formula. In this Note, we find the order of the bias and the variance of the approximation error, and we apply the Malliavin–Thalmaier formula for the calculation of Greeks in finance. To cite this article: A. Kohatsu-Higa, K. Yasuda, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Goodness-of-fit tests via phi-divergences

A unified family of goodness-of-fit tests based on O-divergences is introduced and studied. The new family of test statistics S n (s) includes both the supremum version of the Anderson-Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47-59] as special cases (s = 2 and s = 1, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47-59] and Wellner and Koltchinskii [High Dimensional Probability III (2003) 321-332. Birkhauser, Basel] for the Berk-Jones statistic applies to the whole family of statistics S n (s) with s ∈ [-1, 2]. On the side of power behavior, we study the test statistics under fixed alternatives and give extensions of the Poisson boundary phenomena noted by Berk and Jones for their statistic. We also extend the results of Donoho and Jin [Ann. Statist. 32 (2004) 962-994] by showing that all our new tests for s ∈ [- 1, 2] have the same optimal detection boundary for normal shift mixture alternatives as Tukey's higher-criticism statistic and the Berk-Jones statistic.

Generative and Discriminative Learning by CL-Net

This correspondence presents a two-stage classification learning algorithm. The first stage approximates the class-conditional distribution of a discrete space using a separate mixture model, and the second stage investigates the class posterior probabilities by training a network. The first stage explores the generative information that is inherent in each class by using the Chow-Liu (CL) method, which approximates high-dimensional probability with a tree structure, namely, a dependence tree, whereas the second stage concentrates on discriminative learning to distinguish between classes. The resulting learning algorithm integrates the advantages of both generative learning and discriminative learning. Because it uses CL dependence-tree estimation, we call our algorithm CL-Net. Empirical tests indicate that the proposed learning algorithm makes significant improvements when compared with the related classifiers that are constructed by either generative learning or discriminative learning.

A solver for the stochastic master equation applied to gene regulatory networks

An important driver of gene regulatory networks is noise arising from the stochastic nature of interactions of genes, their products and regulators. Thus, such systems are stochastic and can be modelled by the chemical master equations. A major challenge is the curse of dimensionality which occurs when one attempts to integrate these equations. While stochastic simulation techniques effectively address the curse, many repeated simulations are required to provide precise information about stationary points, bifurcation phenomena and other properties of the stochastic processes. An alternative way to address the curse of dimensionality is provided by sparse grid approximations. The sparse grid methodology is applied and the application demonstrated to work efficiently for up to 10 proteins. As sparse grid methods have been developed for the approximation of smooth functions, a variant for infinite sequences had to be developed together with a multiresolution analysis similar to Haar wavelets. Error bounds are provided which confirm the effectiveness of sparse grid approximations for smooth high-dimensional probability distributions.

Tensor products of convex sets and the volume of separable states onNqudits

This note deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is super-exponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound $6^{-N/2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on N qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces. One relatively non-standard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept. PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Db, 02.40.Ft, 02.50.Cw MSC-class: 46B28, 47B10, 47L05, 52A38, 81P68

Monte Carlo methods for signal processing: a review in the statistical signal processing context

In this article, MCMC (Markov chain Monte Carlo methods) and SMC (sequential Monte Carlo methods) are introduced to sample and/or maximize high-dimensional probability distributions. These methods enable to perform likelihood or Bayesian inference for complex non-Gaussian signal processing problems.

Volume of separable states is super-doubly-exponentially small in the number of qubits

In this paper we give sharp two-sided estimates of the volume of the set of separable states on N qubits. In particular, the magnitude of the 'effective radius' of that set in the sense of volume is determined up to a factor which is a (small) power of N, and thus precisely on the scale of powers of its dimension. We also identify an ellipsoid that appears to optimally approximate the set of separable states. Additionally, one of the appendixes contains sharp estimates (by known methods) for the expected values of norms of the Gaussian unitary ensemble random matrices. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces.

Consistency of the generalized MLE of a joint distribution function with multivariate interval-censored data

Wong and Yu [Generalized MLE of a joint distribution function with multivariate interval-censored data, J. Multivariate Anal. 69 (1999) 155–166] discussed generalized maximum likelihood estimation of the joint distribution function of a multivariate random vector whose coordinates are subject to interval censoring. They established uniform consistency of the generalized MLE (GMLE) of the distribution function under the assumption that the random vector is independent of the censoring vector and that both of the vector distributions are discrete. We relax these assumptions and establish consistency results of the GMLE under a multivariate mixed case interval censorship model. van der Vaart and Wellner [Preservation theorems for Glivenko–Cantelli and uniform Glivenko–Cantelli class, in: E. Gine, D.M. Mason, J.A. Wellner (Eds.), High Dimensional Probability, vol. II, Birkhäuser, Boston, 2000, pp. 115–133] and Yu [Consistency of the generalized MLE with multivariate mixed case interval-censored data, Ph.D Dissertation, Binghamton University, 2000] independently proved strong consistency of the GMLE in the L 1 ( μ ) -topology, where μ is a measure derived from the joint distribution of the censoring variables. We establish strong consistency of the GMLE in the topologies of weak convergence and pointwise convergence, and eventually uniform convergence under appropriate distributional assumptions and regularity conditions.

COMBINATION OF MULTIPLE CLASSIFIERS BY MINIMIZING THE UPPER BOUND OF BAYES ERROR RATE FOR UNCONSTRAINED HANDWRITTEN NUMERAL RECOGNITION

In order to raise a class discrimination power by the combination of multiple classifiers, the upper bound of Bayes error rate which is bounded by the conditional entropy of a class and decisions should be minimized. Based on the minimization of the upper bound of the Bayes error rate, Wang and Wong proposed only a tree dependence approximation scheme of a high-dimensional probability distribution composed of a class and patterns. This paper extends such a tree dependence approximation scheme to higher order dependency for improving the classification performance and thus optimally approximates the high-dimensional probability distribution with a product of low-dimensional distributions. And then, a new combination method by the proposed approximation scheme is presented and evaluated with classifiers recognizing unconstrained handwritten numerals.

On non-linear stochastic dynamics of quarter car models

In this paper, high-dimensional probability density functions of non-linear dynamical systems are calculated solving the corresponding Fokker–Planck equations. Zeroth approximations are derived from solutions of corresponding linear systems and analytical results for first- and second-order expected values. The zeroth approximations are used as weighting functions for the construction of generalized Hermite polynomials. The Fokker–Planck equation is expanded in terms of these polynomials and subsequently solved by a Galerkin method. As an example, models of a quarter car with non-linear damping subjected to white or colored noise excitation are considered. The damping is piecewise linear and asymmetric leading to a non-vanishing expected value of the displacement of the car. The excitation is realized by the roughness of the road and the car moves with constant velocity. Monte-Carlo simulations and analytical results are used for comparison.

Modelling of a turbulent reacting gas/particle flow

A mathematical model for two-phase turbulent reactive flows is presented which is based on considering both phases in Lagrangian manner. The mechanical and thermodynamical properties of the two-phase mixture are calculated along the trajectories of “particles” representing the system. Similar to Monte-Carlo methods for solving a high dimensional joint velocity-composition probability density function, the turbulent gas phase is described by means of stochastic calculus. The deterministic equations for individual solid particles can be treated directly. In this approach, the interaction between both phases is not smeared over computational cells but restricted to the vicinity of solid particles by the definition of an “action-sphere” which is attached to every solid particle. Applications of the method to isotropic, homogeneous turbulence indicate that it is capable of providing information on the local structure of combustion zones with species formation and transport. The results show that the method is applicable independent of the combustion modes in the gas phase, and it provides extensive statistics of various correlations of properties.

Modelling of Turbulent Gas/Particle Combustion by a Lagrangian PDF Method

A mathematical model for two-phase turbulent reactive flows is presented which is based on considering both phases in Lagrangian manner. The mechanical and thermodynamical properties of the two-phase mixture are calculated along the trajectories of ”particles” representing the system. Similar to Monte-Carlo methods for solving a high dimensional joint velocity-composition probability density function, the turbulent gas phase is described by means of stochastic calculus. The deterministic equations for individual solid particles can be treated directly. In this approach, the interaction between both phases is not smeared over computational ceils but restricted to the vicinity of solid particles by the definition of an “action-sphere“ which is attached to every solid particle. Applications of the method to isotropic, homogeneous turbulence indicate that it is capable of providing information on the local structure of combustion zones with species formation and transport. The results show that the method is applicable independent of the combustion modes in the gas phase and it provides extensive statistics of various correlations of properties.

Application of numerical interval analysis to obtain self-validating results for multivariate probabilities in a massively parallel environment

Conventional computations use real numbers as input and produce real numbers as results without any indication of the accuracy. Interval analysis, instead, uses interval elements throughout the computation and produces intervals as output with the guarantee that the true results are contained in them. One major use for interval analysis in statistics is to get results of high-dimensional multivariate probabilities. With the efforts to decrease the length of the intervals that contain the theoretically true answers, we can obtain results to any arbitrary accuracy, which is demonstrated by multivariate normal and multivariate t integrations. This is an advantage over the approximation methods that are currently in use. Since interval analysis is more computationally intensive than traditional computing, a MasPar parallel computer is used in this research to improve performance.

不完全な低次確率データからの高次結合確率密度関数の抽出の一方法

不完全な低次確率データからの高次結合確率密度関数の抽出の一方法

Application of the Statistically Linearized Filter to Re-entry Estimation

THE purpose of this paper is to show that the Statistically Linearized Filter (SLF) is a competitive tool in estimating the of rather complex systems, and to outline its application to reconstructing re-entry trajectories from a variety of sensor data. The SLF is presented as a current-state squareroot filter of the continuous-discrete type. As a crucial test for full-sized application of this algorithm, the estimation of reentry trajectories of ballistic bodies from a variety of sensor data was considered. Both the Extended Kalman Filter (EKF) and SLF were applied to this estimation problem assuming a perfect error model. It was discovered that EKF diverges for this system, whereas SLF performs very well. The cost associated with the evaluation of statistical averages necessary in SLF can be dramatically reduced by computing as many averages as possible on the basis of closed-form expressions using the algebraic properties of the differential equations of motion. SLF was successfully applied to test-flight data employing a minimum error state approach to instrumenterror modeling. Contents Numerous attempts have been made over the years to find nonlinear filters which are workable on complex and realistically sized problems. The SLF has been a prominent candidate because of its formal similarity to the EKF. Examples have shown the power of this method; however, applications have been mostly limited to cases of modest dimension or well-isolated nonlinearities, because the effort in computing statistical averages over high-dimensional probability spaces proved formidable.J The basic formulation of the SLF has been re-examined to remove some of the numerical obstacles blocking the way to more substantial applications. This resulted in an algorithm, for the continuous-discrete case, which employs a square-root decomposition of the covariance matrix, a convenient modification of the Bierman UD-algorithm,2 and the selection of a fifth-order numerical integration scheme for the evaluation of statistical expectations. The SLF can be successfully and economically applied to problems of re-entry estimation. It is assumed that the reentry body carries strap-down instrumentation (bodymounted accelerometers and rate gyroscopes) as well as a magnetometer for attitude determination. The body is tracked by one or several radars and the impact point can be measured very accurately. The motion of the re-entry body involves both translational and rotational degrees of freedom and can

Monte Carlo sampling methods using Markov chains and their applications

SUMMARY A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates. Examples of the methods, including the generation of random orthogonal matrices and potential applications of the methods to numerical problems arising in statistics, are discussed. For numerical problems in a large number of dimensions, Monte Carlo methods are often more efficient than conventional numerical methods. However, implementation of the Monte Carlo methods requires sampling from high dimensional probability distributions and this may be very difficult and expensive in analysis and computer time. General methods for sampling from, or estimating expectations with respect to, such distributions are as follows. (i) If possible, factorize the distribution into the product of one-dimensional conditional distributions from which samples may be obtained. (ii) Use importance sampling, which may also be used for variance reduction. That is, in order to evaluate the integral J = X) p(x)dx = Ev(f), where p(x) is a probability density function, instead of obtaining independent samples XI, ..., Xv from p(x) and using the estimate J, = Zf(xi)/N, we instead obtain the sample from a distribution with density q(x) and use the estimate J2 = Y{f(xj)p(x1)}/{q(xj)N}. This may be advantageous if it is easier to sample from q(x) thanp(x), but it is a difficult method to use in a large number of dimensions, since the values of the weights w(xi) = p(x1)/q(xj) for reasonable values of N may all be extremely small, or a few may be extremely large. In estimating the probability of an event A, however, these difficulties may not be as serious since the only values of w(x) which are important are those for which x -A. Since the methods proposed by Trotter & Tukey (1956) for the estimation of conditional expectations require the use of importance sampling, the same difficulties may be encountered in their use. (iii) Use a simulation technique; that is, if it is difficult to sample directly from p(x) or if p(x) is unknown, sample from some distribution q(y) and obtain the sample x values as some function of the corresponding y values. If we want samples from the conditional dis