Estimating mutual information from i.i.d. samples drawn from an unknown joint density function is a basic statistical problem of broad interest with multitudinous applications. The most popular estimator is one proposed by Kraskov and St "ogbauer and Grassberger (KSG) in 2004, and is nonparametric and based on the distances of each sample to its @math nearest neighboring sample, where @math is a fixed small integer. Despite its widespread use (part of scientific software packages), theoretical properties of this estimator have been largely unexplored. In this paper we demonstrate that the estimator is consistent and also identify an upper bound on the rate of convergence of the bias as a function of number of samples. We argue that the superior performance benefits of the KSG estimator stems from a curious "correlation boosting" effect and build on this intuition to modify the KSG estimator in novel ways to construct a superior estimator. As a byproduct of our investigations, we obtain nearly tight rates of convergence of the @math error of the well known fixed @math nearest neighbor estimator of differential entropy by Kozachenko and Leonenko.
In some cases, the parametric rate of convergence of @math of @math error is achieved: of particular interest is the result of @cite_60 where the parametric rate is achieved for differential entropy estimation via KDE of density followed by the resubstitution estimator when the dimension is no more than 6. Numerical evidence suggests the hypothesis that the lower bounds derived in Theorem below could perhaps be improved when the dimension is more than 4 and estimators constrained to only use fixed @math -NN distances. Under certain very strong conditions on the density class (that are relevant in certain applications on graphical model selection @cite_55 ), exponential rate of convergence can be demonstrated @cite_66 @cite_33 . Recent works @cite_63 @cite_0 have studied the performance of the leave-one-out (LOO) approach where all but the sample of resubstitution are used for KDE, involving techniques such as von Mises expansion methods.
@cite_33 @cite_60 @cite_55 @cite_0 @cite_63 @cite_66
2160184181 2029140344 2143908925 2188155116 1673728053 2963540279
We derive the mean squared error convergence rates of kernel density-based plug-in estimators of mutual information measures between two multidimensional random variables @math and @math for two cases: 1) @math and @math are both continuous; 2) @math is continuous and @math is discrete. Using the derived rates, we propose an ensemble estimator of these information measures for the second case by taking a weighted sum of the plug-in estimators with varied bandwidths. The resulting ensemble estimator achieves the @math parametric convergence rate when the conditional densities of the continuous variables are sufficiently smooth. To the best of our knowledge, this is the first nonparametric mutual information estimator known to achieve the parametric convergence rate for this case, which frequently arises in applications (e.g. variable selection in classification). The estimator is simple to implement as it uses the solution to an offline convex optimization problem and simple plug-in estimators. A central limit theorem is also derived for the ensemble estimator. Ensemble estimators that achieve the parametric rate are also derived for the first case ( @math and @math are both continuous) and another case 3) @math and @math may have any mixture of discrete and continuous components. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results. We consider nonparametric estimation of L2, Renyi- and Tsallis- divergences between continuous distributions. Our approach is to construct estimators for particular integral functionals of two densities and translate them into divergence estimators. For the integral functionals, our estimators are based on corrections of a preliminary plug-in estimator. We show that these estimators achieve the parametric convergence rate of n-1 2 when the densities' smoothness, s, are both at least d 4 where d is the dimension. We also derive minimax lower bounds for this problem which confirm that s > d 4 is necessary to achieve the n-1 2 rate of convergence. We validate our theoretical guarantees with a number of simulations. Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case, the least-squares method is quasi-optimal in expectation in [A. Cohen, M A. Davenport and D. Leviatan. Found. Comput. Math. 13 (2013) 819834] and in probability in [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Found. Comput. Math. 14 (2014) 419456], under suitable conditions that relate the number of samples with respect to the dimension of the polynomial space. Here quasi-optimal means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the quasi-optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space, independently of the anisotropic shape and of the number of variables. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss inclusion type elliptic PDE models, and derive an exponential convergence estimate for the least-squares method. Numerical results confirm our estimate, yet pointing out a gap b Sampling distribution over high-dimensional state-space is a problem which has recently attracted a lot of research efforts; applications include Bayesian non-parametrics, Bayesian inverse problems and aggregation of estimators. All these problems boil down to sample a target distribution @math having a density the Lebesgue measure on @math , known up to a normalisation factor @math where @math is continuously differentiable and smooth. In this paper, we study a sampling technique based on the Euler discretization of the Langevin stochastic differential equation. Contrary to the Metropolis Adjusted Langevin Algorithm (MALA), we do not apply a Metropolis-Hastings correction. We obtain for both constant and decreasing step sizes in the Euler discretization, non-asymptotic bounds for the convergence to stationarity in both total variation and Wasserstein distances. A particular attention is paid on the dependence on the dimension of the state space, to demonstrate the applicability of this method in the high dimensional setting, at least when @math is convex. These bounds are based on recently obtained estimates of the convergence of the Langevin diffusion to stationarity using Poincar 'e and log-Sobolev inequalities. These bounds improve and extend the results of (Dalalyan, 2014). We also investigate the convergence of an appropriately weighted empirical measure and we report sharp bounds for the mean square error and exponential deviation inequality for Lipschitz functions. A limited Monte Carlo experiment is carried out to support our findings.