Showing papers on "Measure (mathematics) published in 2009"

A kernel statistical test of independence

Abstract: Although kernel measures of independence have been widely applied in machine learning (notably in kernel ICA), there is as yet no method to determine whether they have detected statistically significant dependence. We provide a novel test of the independence hypothesis for one particular kernel independence measure, the Hilbert-Schmidt independence criterion (HSIC). The resulting test costs O(m2), wherem is the sample size. We demonstrate that this test outperforms established contingency table and functional correlation-based tests, and that this advantage is greater for multivariate data. Finally, we show the HSIC test also applies to text (and to structured data more generally), for which no other independence test presently exists.

A Price Theory of Multi-Sided Platforms

Abstract: I develop a general theory of monopoly pricing of networks. Platforms use insulating tariffs to avoid coordination failure, implementing any desired allocation. Profit-maximization distorts in the spirit of Spence (1975) by internalizing only network externalities to marginal users. Thus the empirical and prescriptive content of the popular Rochet and Tirole (2006) model of two-sided markets turns on the nature of user heterogeneity. I propose a more plausible, yet equally tractable, model of heterogeneity in which users differ in their income or scale. My approach provides a general measure of market power and helps predict the effects of price regulation and mergers.

Non-iterative approach for fast and accurate vanishing point detection

Abstract: We present an algorithm that quickly and accurately estimates vanishing points in images of man-made environments. Contrary to previously proposed solutions, ours is neither iterative nor relies on voting in the space of vanishing points. Our formulation is based on a recently proposed algorithm for the simultaneous estimation of multiple models called J-Linkage. Our method avoids representing edges on the Gaussian sphere and the computations and error measures are done in the image. We show that a consistency measure between a vanishing point and an edge of the image can be computed in closed-form while being geometrically meaningful. Finally, given a set of estimated vanishing points, we show how this consistency measure can be used to identify the three vanishing points corresponding to the Manhattan directions. We compare our algorithm with other approaches on the York Urban Database and show significant performance improvements.

The Calderón-Zygmund theory for elliptic problems with measure data

Abstract: We consider non-linear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = μ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderon-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates. To the memory of Vic Mizel, mathematician and gentleman

A measure & conquer approach for the analysis of exact algorithms

Abstract: For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call “Measure & Conquer”, as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis.In order to show the potentialities of Measure & Conquer, we consider two well-studied NP-hard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis).Our examples show that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.

Comparison of Pedestrian Fundamental Diagram Across Cultures

Abstract: The relation between speed and density is connected with every self-organization phenomenon of pedestrian dynamics and offers the opportunity to analyze them quantitatively. But even for the simplest systems, like pedestrian streams in corridors, this fundamental relation isn't completely understood. Specifications of this characteristic in guidelines and text books differ considerably reflecting the contradictory database and the controversial discussion documented in the literature. In this contribution it is studied whether cultural influences and length of the corridor can be the causes for these deviations. To reduce as much as possible unintentioned effects, a system is chosen with reduced degrees of freedom and thus the most simple system, namely the movement of pedestrians along a line under closed boundary conditions. It is found that the speed of Indian test persons is less dependent on density than the speed of German test persons. Surprisingly the more unordered behaviour of the Indians is more effective than the ordered behaviour of the Germans. Without any statistical measure one cannot conclude about whether there are differences or not. By hypothesis test it is found quantitatively that these differences exist, suggesting cultural differences in the fundamental diagram of pedestrians.

Bayesian inverse problems for functions and applications to fluid mechanics

Abstract: In this paper we establish a mathematical framework for a range of inverse problems for functions, given a finite set of noisy observations. The problems are hence underdetermined and are often ill-posed. We study these problems from the viewpoint of Bayesian statistics, with the resulting posterior probability measure being defined on a space of functions. We develop an abstract framework for such problems which facilitates application of an infinite-dimensional version of Bayes theorem, leads to a well-posedness result for the posterior measure (continuity in a suitable probability metric with respect to changes in data), and also leads to a theory for the existence of maximizing the posterior probability (MAP) estimators for such Bayesian inverse problems on function space. A central idea underlying these results is that continuity properties and bounds on the forward model guide the choice of the prior measure for the inverse problem, leading to the desired results on well-posedness and MAP estimators; the PDE analysis and probability theory required are thus clearly dileneated, allowing a straightforward derivation of results. We show that the abstract theory applies to some concrete applications of interest by studying problems arising from data assimilation in fluid mechanics. The objective is to make inference about the underlying velocity field, on the basis of either Eulerian or Lagrangian observations. We study problems without model error, in which case the inference is on the initial condition, and problems with model error in which case the inference is on the initial condition and on the driving noise process or, equivalently, on the entire time-dependent velocity field. In order to undertake a relatively uncluttered mathematical analysis we consider the two-dimensional Navier–Stokes equation on a torus. The case of Eulerian observations—direct observations of the velocity field itself—is then a model for weather forecasting. The case of Lagrangian observations—observations of passive tracers advected by the flow—is then a model for data arising in oceanography. The methodology which we describe herein may be applied to many other inverse problems in which it is of interest to find, given observations, an infinite-dimensional object, such as the initial condition for a PDE. A similar approach might be adopted, for example, to determine an appropriate mathematical setting for the inverse problem of determining an unknown tensor arising in a constitutive law for a PDE, given observations of the solution. The paper is structured so that the abstract theory can be read independently of the particular problems in fluid mechanics which are subsequently studied by application of the theory.

Measuring Morphological Integration Using Eigenvalue Variance

Abstract: The concept of morphological integration describes the pattern and the amount of correlation between morphological traits. Integration is relevant in evolutionary biology as it imposes constraint on the variation that is exposed to selection, and is at the same time often based on heritable genetic correlations. Several measures have been proposed to assess the amount of integration, many using the distribution of eigenvalues of the correlation matrix. In this paper, we analyze the properties of eigenvalue variance as a much applied measure. We show that eigenvalue variance scales linearly with the square of the mean correlation and propose the standard deviation of the eigenvalues as a suitable alternative that scales linearly with the correlation. We furthermore develop a relative measure that is independent of the number of traits and can thus be readily compared across datasets. We apply this measure to examples of phenotypic correlation matrices and compare our measure to several other methods. The relative standard deviation of the eigenvalues gives similar results as the mean absolute correlation (W.P. Cane, Evol Int J Org Evol 47:844–854, 1993) but is only identical to this measure if the correlation matrix is homogenous. For heterogeneous correlation matrices the mean absolute correlation is consistently smaller than the relative standard deviation of eigenvalues and may thus underestimate integration. Unequal allocation of variance due to variation among correlation coefficients is captured by the relative standard deviation of eigenvalues. We thus suggest that this measure is a better reflection of the overall morphological integration than the average correlation.

A new class of transport distances between measures

Abstract: We introduce a new class of distances between nonnegative Radon measures in \({\mathbb{R}^d}\) . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous \({W^{-1,p}_\gamma}\) -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

S-metric calculation by considering dominated hypervolume as klee's measure problem

Abstract: The dominated hypervolume (or S-metric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multi-objective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the S-metric internally several times per iteration, a fast determination of the S-metric value is of essential importance. This work describes how to consider the S-metric as a special case of a more general geometric problem called Klee's measure problem (KMP). For KMP, an algorithm exists with runtime O(n log n + nd/2 log n), for n points of d ≥ 3 dimensions. This complex algorithm is adapted to the special case of calculating the S-metric. Conceptual simplifications realize the algorithm without complex data structures and establish an upper bound of O(nd/2 log n) for the S-metric calculation for d ≥ 3. The performance of the new algorithm is studied in comparison to another state of the art algorithm on a set of academic test functions.

Some properties of continuous uncertain measure

Abstract: In this paper, we discuss some properties in uncertainty theory when uncertain measure is continuous. Firstly, the judgement conditions of continuous uncertain measure are proposed. Secondly, basic properties of uncertainty distribution and critical values of uncertain variable are proved. Finally, the convergence theorems for expected value are discussed.

Clustering for Metric and Non-Metric Distance Measures ⁄ (full version)

Abstract: We study a generalization of the k-median problem with respect to an arbitrary dissimilarity measure D. Given a finite set P of size n, our goal is to find a set C of size k such that the sum of errors D(P,C) = P p2P minc2C ' D(p,c) “ is minimized. The main result in this paper can be stated as follows: There exists a (1+†)-approximation algorithm for the k-median problem with respect to D, if the 1-median problem can be approximated within a factor of (1+†) by taking a random sample of constant size and solving the 1-median problem on the sample exactly. This algorithms requires time n2 O(mk log(mk/†)) , where m is a constant that depends only on † and D. Using this characterization, we obtain the first linear time (1+†)-approximation algorithms for the k-median problem in an arbitrary metric space with bounded doubling dimension, for the Kullback-Leibler divergence (relative entropy), for the Itakura-Saito divergence, for Mahalanobis distances, and for some special cases of Bregman divergences. Moreover, we obtain previously known results for the Euclidean k-median problem and the Euclidean k-means problem in a simplified manner. Our results are based on a new analysis of an algorithm of Kumar, Sabharwal, and Sen from FOCS 2004.

Matrix-Valued Szegő Polynomials and Quantum Random Walks

Abstract: We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case, the study of QRWs on the set of nonnegative integers can be handled using scalar-valued (Laurent) polynomials and a scalar-valued measure on the circle. In the case of classical or quantum random walks on the integers, one needs to allow for matrix-valued versions of these notions. We show how our tools yield results in the well-known case of the Hadamard walk, but we go beyond this translation-invariant model to analyze examples that are hard to analyze using other methods. More precisely, we consider QRWs on the set of nonnegative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being self-contained, but we refrain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 9]. See also the recent notes [20]. © 2009 Wiley Periodicals, Inc.

Properties of the scale factor measure

Abstract: We show that in expanding regions, the scale factor measure can be reformulated as a local measure: Observations are weighted by integrating their physical density along a geodesic that starts in the longest-lived metastable vacuum. This explains why some of its properties are similar to those of the causal-diamond measure. In particular, both measures are free of Boltzmann brains, subject to nearly the same conditions on vacuum stability. However, the scale factor measure assigns a much smaller probability to the observed value of the cosmological constant. The probability decreases further, similar to the inverse sixth power of the primordial density contrast, if the latter is allowed to vary.

Optimal rates for plug-in estimators of density level sets

Abstract: In the context of density level set estimation, we study the convergence of general plug-in methods under two main assumptions on the density for a given level λ. More precisely, it is assumed that the density (i) is smooth in a neighborhood of λ and (ii) has γ -exponent at level λ. Condition (i) ensures that the density can be estimated at a standard nonparametric rate and condition (ii) is similar to Tsybakov’s margin assumption which is stated for the classification framework. Under these assumptions, we derive optimal rates of convergence for plug-in estimators. Explicit convergence rates are given for plug-in estimators based on kernel density estimators when the underlying measure is the Lebesgue measure. Lower bounds proving optimality of the rates in a minimax sense when the density is Holder smooth are also provided.

Matrix Representation of the Stationary Measure for the Multispecies TASEP

Abstract: In this work we construct the stationary measure of the N species totally asymmetric simple exclusion process in a matrix product formulation. We make the connection between the matrix product formulation and the queueing theory picture of Ferrari and Martin. In particular, in the standard representation, the matrices act on the space of queue lengths. For N>2 the matrices in fact become tensor products of elements of quadratic algebras. This enables us to give a purely algebraic proof of the stationary measure which we present for N=3.

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Abstract: Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ℝ3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ e ≤ 1, |v (x, t)| ≤ C ∗ r −1+e |t|−e/2 for − T 0 ≤ t < 0 and 0 < C ∗ < ∞ allowed to be large. We prove that v is regular at time zero.

Lobby index in networks

Abstract: We propose a new node centrality measure in networks, the lobby index, which is inspired by Hirsch’s h -index. It is shown that in scale-free networks with exponent α the distribution of the l -index has power tail with exponent α ( α + 1 ) . Properties of the l -index and extensions are discussed.

High-precision predictions for the acoustic scale in the non-linear regime

Abstract: We measure shifts of the acoustic scale due to nonlinear growth and redshift distortions to a high precision using a very large volume of high-force-resolution simulations. We compare results from various sets of simulations that differ in their force, volume, and mass resolution. We find a consistency within 1.5-sigma for shift values from different simulations and derive shift alpha(z) -1 = (0.300\pm 0.015)% [D(z)/D(0)]^{2} using our fiducial set. We find a strong correlation with a non-unity slope between shifts in real space and in redshift space and a weak correlation between the initial redshift and low redshift. Density-field reconstruction not only removes the mean shifts and reduces errors on the mean, but also tightens the correlations: after reconstruction, we recover a slope of near unity for the correlation between the real and redshift space and restore a strong correlation between the low and the initial redshifts. We derive propagators and mode-coupling terms from our N-body simulations and compared with Zeldovich approximation and the shifts measured from the chi^2 fitting, respectively. We interpret the propagator and the mode-coupling term of a nonlinear density field in the context of an average and a dispersion of its complex Fourier coefficients relative to those of the linear density field; from these two terms, we derive a signal-to-noise ratio of the acoustic peak measurement. We attempt to improve our reconstruction method by implementing 2LPT and iterative operations: we obtain little improvement. The Fisher matrix estimates of uncertainty in the acoustic scale is tested using 5000 (Gpc/h)^3 of cosmological PM simulations from Takahashi et al. (2009). (abridged)

Local stability of ergodic averages

Abstract: We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of [0,1] and a characteristic function f = XA such that the ergodic averages A n f do not converge to a computable element of L 2 ([0, 1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T on a separable Hilbert space and any element f, it is possible to compute a bound on the rate of convergence of 〈A n f〉 from T, f, and the norm ∥f*∥ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L 2 (X), then there is a computable bound on the rate of convergence of the sequence 〈Anf〉. The mean ergodic theorem is equivalent to the assertion that for every function K(n) and every e > 0, there is an n with the property that the ergodic averages A m f are stable to within e on the interval [n, K(n)]. Even in situations where the sequence 〈A n f〉 does not have a computable limit, one can give explicit bounds on such n in terms of K and ∥f∥/e. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining.".

Weighted poincaré-type inequalities for cauchy and other convex measures

Abstract: Brascamp-Lieb-type, weighted Poincare-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.

A new class of models for bivariate joint tails

Abstract: Summary. A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.

On the local optimality of LambdaRank

Abstract: A machine learning approach to learning to rank trains a model to optimize a target evaluation measure with repect to training data. Currently, existing information retrieval measures are impossible to optimize directly except for models with a very small number of parameters. The IR community thus faces a major challenge: how to optimize IR measures of interest directly. In this paper, we present a solution. Specifically, we show that LambdaRank, which smoothly approximates the gradient of the target measure, can be adapted to work with four popular IR target evaluation measures using the same underlying gradient construction. It is likely, therefore, that this construction is extendable to other evaluation measures. We empirically show that LambdaRank finds a locally optimal solution for mean NDCG@10, mean NDCG, MAP and MRR with a 99% confidence rate. We also show that the amount of effective training data varies with IR measure and that with a sufficiently large training set size, matching the training optimization measure to the target evaluation measure yields the best accuracy.

Validation of imprecise probability models

Abstract: Validation is the assessment of the match between a model's predictions and empirical observations. It can be complex when either data or the prediction is characterised as an uncertain number (i.e. interval, probability distribution, p-box, or more general structure). Validation could measure the discrepancy between the shapes of the two uncertain numbers representing prediction and data, or it could characterise the differences between realisations drawn from the respective uncertain numbers. The unification between these two concepts relies on defining the validation measure between prediction and data as the shortest possible distance given the imprecision about the distributions and their dependencies.