Showing papers on "Gaussian published in 2009"
TL;DR: This work analyzes the behavior of l1-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern of a vector beta* based on observations contaminated by noise, and establishes precise conditions on the problem dimension p, the number k of nonzero elements in beta*, and the number of observations n.
Abstract: The problem of consistently estimating the sparsity pattern of a vector beta* isin Rp based on observations contaminated by noise arises in various contexts, including signal denoising, sparse approximation, compressed sensing, and model selection. We analyze the behavior of l1-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern. Our main result is to establish precise conditions on the problem dimension p, the number k of nonzero elements in beta*, and the number of observations n that are necessary and sufficient for sparsity pattern recovery using the Lasso. We first analyze the case of observations made using deterministic design matrices and sub-Gaussian additive noise, and provide sufficient conditions for support recovery and linfin-error bounds, as well as results showing the necessity of incoherence and bounds on the minimum value. We then turn to the case of random designs, in which each row of the design is drawn from a N (0, Sigma) ensemble. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we compute explicit values of thresholds 0 0, if n > 2 (thetasu + delta) klog (p- k), then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for n < 2 (thetasl - delta)klog (p - k), then the probability of successful recovery converges to zero. For the special case of the uniform Gaussian ensemble (Sigma = Iptimesp), we show that thetasl = thetas
1,438 citations
01 Jan 2009
TL;DR: Gaussian Mixture Model parameters are estimated from training data using the iterative Expectation-Maximization (EM) algorithm or Maximum A Posteriori (MAP) estimation from a well-trained prior model.
Abstract: Definition A Gaussian Mixture Model (GMM) is a parametric probability density function represented as a weighted sum of Gaussian component densities. GMMs are commonly used as a parametric model of the probability distribution of continuous measurements or features in a biometric system, such as vocal-tract related spectral features in a speaker recognition system. GMM parameters are estimated from training data using the iterative Expectation-Maximization (EM) algorithm or Maximum A Posteriori (MAP) estimation from a well-trained prior model.
1,323 citations
TL;DR: In this paper, the perturbation approach originally introduced by Moller and Plesset, terminated at finite order, is compared from the point of view of requirements for theoretical chemical models.
Abstract: Some methods of describing electron correlation are compared from the point of view of requirements for theoretical chemical models. The perturbation approach originally introduced by Moller and Plesset, terminated at finite order, is found to satisfy most of these requirements. It is size consistent, that is, applicable to an ensemble of isolated systems in an additive manner. On the other hand, it does not provide an upper bound for the electronic energy. The independent electron-pair approximation is accurate to second order in a Moller-Plesset expansion, but inaccurate in third order. A series of variational methods is discussed which gives upper bounds for the energy, but which lacks size consistency. Finally, calculations on some small molecules using a moderately large Gaussian basis are presented to illustrate these points. Equilibrium geometries, dissociation energies, and energy separations between electronic states of different spin multiplicities are describe substantially better by Moller--Plesset theory to second or third order than by Hartree--Fock theory.
1,217 citations
TL;DR: It is shown analytically that the multitarget multiBernoulli (MeMBer) recursion, proposed by Mahler, has a significant bias in the number of targets and to reduce the cardinality bias, a novel multi Bernoulli approximation to the multi-target Bayes recursion is derived.
Abstract: It is shown analytically that the multitarget multiBernoulli (MeMBer) recursion, proposed by Mahler, has a significant bias in the number of targets. To reduce the cardinality bias, a novel multiBernoulli approximation to the multi-target Bayes recursion is derived. Under the same assumptions as the MeMBer recursion, the proposed recursion is unbiased. In addition, a sequential Monte Carlo (SMC) implementation (for generic models) and a Gaussian mixture (GM) implementation (for linear Gaussian models) are proposed. The latter is also extended to accommodate mildly nonlinear models by linearization and the unscented transform.
741 citations
TL;DR: In this paper, a deterministic channel model was proposed for Gaussian networks with a single source and a single destination and an arbitrary number of relay nodes, and a quantize-map-and-forward scheme was proposed.
Abstract: In a wireless network with a single source and a single destination and an arbitrary number of relay nodes, what is the maximum rate of information flow achievable? We make progress on this long standing problem through a two-step approach. First we propose a deterministic channel model which captures the key wireless properties of signal strength, broadcast and superposition. We obtain an exact characterization of the capacity of a network with nodes connected by such deterministic channels. This result is a natural generalization of the celebrated max-flow min-cut theorem for wired networks. Second, we use the insights obtained from the deterministic analysis to design a new quantize-map-and-forward scheme for Gaussian networks. In this scheme, each relay quantizes the received signal at the noise level and maps it to a random Gaussian codeword for forwarding, and the final destination decodes the source's message based on the received signal. We show that, in contrast to existing schemes, this scheme can achieve the cut-set upper bound to within a gap which is independent of the channel parameters. In the case of the relay channel with a single relay as well as the two-relay Gaussian diamond network, the gap is 1 bit/s/Hz. Moreover, the scheme is universal in the sense that the relays need no knowledge of the values of the channel parameters to (approximately) achieve the rate supportable by the network. We also present extensions of the results to multicast networks, half-duplex networks and ergodic networks.
601 citations
27 Feb 2009
TL;DR: In this article, the authors present a generalization of the Rice series for Gaussian processes with continuous paths and show that it is invariant under orthogonal transformations and translations.
Abstract: Introduction. Reading diagram. Chapter 1: Classical results on the regularity of the paths. 1. Kolmogorov's Extension Theorem. 2. Reminder on the Normal Distribution. 3. 0-1 law for Gaussian processes. 4. Regularity of the paths. Exercises. Chapter 2: Basic Inequalities for Gaussian Processes. 1. Slepian type inequalities. 2. Ehrhard's inequality. 3. Gaussian isoperimetric inequality. 4. Inequalities for the tails of the distribution of the supremum. 5. Dudley's inequality. Exercises. Chapter 3: Crossings and Rice formulas for 1-dimensional parameter processes. 1. Rice Formulas. 2. Variants and Examples. Exercises. Chapter 4: Some Statistical Applications. 1. Elementary bounds for P{M > u}. 2. More detailed computation of the first two moments. 3. Maximum of the absolute value. 4. Application to quantitative gene detection. 5. Mixtures of Gaussian distributions. Exercises. Chapter 5: The Rice Series. 1. The Rice Series. 2. Computation of Moments. 3. Numerical aspects of Rice Series. 4. Processes with Continuous Paths. Chapter 6: Rice formulas for random fields. 1. Random fields from Rd to Rd. 2. Random fields from Rd to Rd!, d> d!. Exercises. Chapter 7: Regularity of the Distribution of the Maximum. 1. The implicit formula for the density of the maximum. 2. One parameter processes. 3. Continuity of the density of the maximum of random fields. Exercises. Chapter 8: The tail of the distribution of the maximum. 1. One-dimensional parameter: asymptotic behavior of the derivatives of FM. 2. An Application to Unbounded Processes. 3. A general bound for pM. 4. Computing p(x) for stationary isotropic Gaussian fields. 5. Asymptotics as x! +". 6. Examples. Exercises. Chapter 9: The record method. 1. Smooth processes with one dimensional parameter. 2. Non-smooth Gaussian processes. 3. Two-parameter Gaussian processes. Exercises. Chapter 10: Asymptotic methods for infinite time horizon. 1. Poisson character of "high" up-crossings. 2. Central limit theorem for non-linear functionals. Exercises. Chapter 11: Geometric characteristics of random sea-waves. 1. Gaussian model for infinitely deep sea. 2. Some geometric characteristics of waves. 3. Level curves, crests and velocities for space waves. 4. Real Data. 5. Generalizations of the Gaussian model. Exercises. Chapter 12: Systems of random equations. 1. The Shub-Smale model. 2. More general models. 3. Non-centered systems (smoothed analysis). 4. Systems having a law invariant under orthogonal transformations and translations. Chapter 13: Random fields and condition numbers of random matrices. 1. Condition numbers of non-Gaussian matrices. 2. Condition numbers of centered Gaussian matrices. 3. Non-centered Gaussian matrices. Notations. References.
578 citations
TL;DR: It is proved that the full Han-Kobayashi achievable rate region using Gaussian codebooks is equivalent to that of the one-sided Gaussian IC for a particular range of channel parameters.
Abstract: The capacity region of the two-user Gaussian interference channel (IC) is studied. Three classes of channels are considered: weak, one-sided, and mixed Gaussian ICs. For the weak Gaussian IC, a new outer bound on the capacity region is obtained that outperforms previously known outer bounds. The sum capacity for a certain range of channel parameters is derived. For this range, it is proved that using Gaussian codebooks and treating interference as noise are optimal. It is shown that when Gaussian codebooks are used, the full Han-Kobayashi achievable rate region can be obtained by using the naive Han-Kobayashi achievable scheme over three frequency bands (equivalently, three subspaces). For the one-sided Gaussian IC, an alternative proof for the Sato's outer bound is presented. We derive the full Han-Kobayashi achievable rate region when Gaussian codebooks are utilized. For the mixed Gaussian IC, a new outer bound is obtained that outperforms previously known outer bounds. For this case, the sum capacity for the entire range of channel parameters is derived. It is proved that the full Han-Kobayashi achievable rate region using Gaussian codebooks is equivalent to that of the one-sided Gaussian IC for a particular range of channel parameters.
567 citations
TL;DR: In this paper, the authors combine Malliavin calculus with Stein's method to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process.
Abstract: We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener–Ito integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry–Esseen bounds in the Breuer–Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler’s formula for Ornstein–Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.
473 citations
TL;DR: In this short review, the correct and general forms of the Gaussian equation and the approximation of the CD curve by other methods are described.
462 citations
TL;DR: A new region-based active contour model in a variational level set formulation for image segmentation that is able to distinguish regions with similar intensity means but different variances and is demonstrated by applying the method on noisy and texture images.
TL;DR: It is shown that in such a way it is straightforward to combine calculation of Franck-Condon integrals with any electronic computational model.
Abstract: A general and effective time-independent approach to compute vibrationally resolved electronic spectra from first principles has been integrated into the Gaussian computational chemistry package. This computational tool offers a simple and easy-to-use way to compute theoretical spectra starting from geometry optimization and frequency calculations for each electronic state. It is shown that in such a way it is straightforward to combine calculation of Franck-Condon integrals with any electronic computational model. The given examples illustrate the calculation of absorption and emission spectra, all in the UV-vis region, of various systems from small molecules to large ones, in gas as well as in condensed phases. The computational models applied range from fully quantum mechanical descriptions to discrete/continuum quantum mechanical/molecular mechanical/polarizable continuum models.
TL;DR: The relationship between the Laplace and the variational approximation is discussed, and it is shown that for models with gaussian priors and factorizing likelihoods, the number of variational parameters is actually .
Abstract: The variational approximation of posterior distributions by multivariate gaussians has been much less popular in the machine learning community compared to the corresponding approximation by factorizing distributions. This is for a good reason: the gaussian approximation is in general plagued by an number of variational parameters to be optimized, being the number of random variables. In this letter, we discuss the relationship between the Laplace and the variational approximation, and we show that for models with gaussian priors and factorizing likelihoods, the number of variational parameters is actually . The approach is applied to gaussian process regression with nongaussian likelihoods.
TL;DR: It is shown that, for this channel, Gaussian signalling in the form of beam-forming is optimal, and no pre-processing of information is necessary.
Abstract: We find the secrecy capacity of the 2-2-1 Gaussian MIMO wiretap channel, which consists of a transmitter and a receiver with two antennas each, and an eavesdropper with a single antenna. We determine the secrecy capacity of this channel by proposing an achievable scheme and then developing a tight upper bound that meets the proposed achievable secrecy rate. We show that, for this channel, Gaussian signalling in the form of beam-forming is optimal, and no pre-processing of information is necessary.
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TL;DR: This paper proposes an achievable scheme composed of nested lattice codes for the uplink and structured binning for the downlink and shows that the scheme achieves within 1/2 bit from the cut-set bound for all channel parameters and becomes asymptotically optimal as the signal to noise ratios increase.
Abstract: In this paper, a Gaussian two-way relay channel, where two source nodes exchange messages with each other through a relay, is considered. We assume that all nodes operate in full-duplex mode and there is no direct channel between the source nodes. We propose an achievable scheme composed of nested lattice codes for the uplink and structured binning for the downlink. We show that the scheme achieves within 1/2 bit from the cut-set bound for all channel parameters and becomes asymptotically optimal as the signal to noise ratios increase.
TL;DR: It is proved that Gaussian operations are of no use for protecting Gaussian states against Gaussian errors in quantum communication protocols, and a new quantity is introduced characterizing any single-mode Gaussian channel, called entanglement degradation, and it cannot decrease via Gaussian encoding and decoding operations only.
Abstract: We prove that Gaussian operations are of no use for protecting Gaussian states against Gaussian errors in quantum communication protocols. Specifically, we introduce a new quantity characterizing any single-mode Gaussian channel, called entanglement degradation, and show that it cannot decrease via Gaussian encoding and decoding operations only. The strength of this no-go theorem is illustrated with some examples of Gaussian channels.
01 Jan 2009
TL;DR: This chapter discusses the basic notions about state space models and their use in time series analysis, and the dynamic linear model is presented as a special case of a general state space model, being linear and Gaussian.
Abstract: In this chapter we discuss the basic notions about state space models and their use in time series analysis. The dynamic linear model is presented as a special case of a general state space model, being linear and Gaussian. For dynamic linear models, estimation and forecasting can be obtained recursively by the well-known Kalman filter.
TL;DR: The modified Gaussian window is found to provide excellent normalized frequency contours of the power signal disturbances suitable for accurate detection, localization, and classification of various nonstationary power signals.
Abstract: This paper presents a new approach for the visual localization, detection, and classification of various nonstationary power signals using a variety of windowing techniques. Among the various windows used earlier like sine, cosine, tangent, hyperbolic tangent, Gaussian, bi-Gaussian, and complex, the modified Gaussian window is found to provide excellent normalized frequency contours of the power signal disturbances suitable for accurate detection, localization, and classification. Various nonstationary power signals are processed through the generalized S-transform with modified Gaussian window to generate time-frequency contours for extracting relevant features for pattern classification. The extracted features are clustered using fuzzy C-means algorithm, and finally, the algorithm is extended using either particle swarm optimization or genetic algorithm to refine the cluster centers.
TL;DR: In this article, a Gaussian mixture probability hypothesis density (GM-PHD) recursion is proposed for jointly estimating the time-varying number of targets and their states from a sequence of noisy measurement sets.
Abstract: The Gaussian mixture probability hypothesis density (GM-PHD) recursion is a closed-form solution to the probability hypothesis density (PHD) recursion, which was proposed for jointly estimating the time-varying number of targets and their states from a sequence of noisy measurement sets in the presence of data association uncertainty, clutter, and miss-detection. However the GM-PHD filter does not provide identities of individual target state estimates, that are needed to construct tracks of individual targets. In this paper, we propose a new multi-target tracker based on the GM-PHD filter, which gives the association amongst state estimates of targets over time and provides track labels. Various issues regarding initiating, propagating and terminating tracks are discussed. Furthermore, we also propose a technique for resolving identities of targets in close proximity, which the PHD filter is unable to do on its own.
TL;DR: This work has found that standard hybrid functionals can be transformed into short-range functionals without loss of accuracy in Gaussian basis sets, leading to a stable and accurate procedure for evaluating Hartree-Fock exchange at the Γ-point.
Abstract: Hartree-Fock exchange with a truncated Coulomb operator has recently been discussed in the context of periodic plane-waves calculations [Spencer, J.; Alavi, A. Phys. Rev. B: Solid State, 2008, 77, 193110]. In this work, this approach is extended to Gaussian basis sets, leading to a stable and accurate procedure for evaluating Hartree-Fock exchange at the Γ-point. Furthermore, it has been found that standard hybrid functionals can be transformed into short-range functionals without loss of accuracy. The well-defined short-range nature of the truncated exchange operator can naturally be exploited in integral screening procedures and makes this approach interesting for both condensed phase and gas phase systems. The presented Hartree-Fock implementation is massively parallel and scales up to ten thousands of cores. This makes it feasible to perform highly accurate calculations on systems containing thousands of atoms or ten thousands of basis functions. The applicability of this scheme is demonstrated by calculating the cohesive energy of a LiH crystal close to the Hartree-Fock basis set limit and by performing an electronic structure calculation of a complete protein (rubredoxin) in solution with a large and flexible basis set.
01 Jan 2009
TL;DR: Gaussian processes are in my view the simplest and most obvious way of defining flexible Bayesian regression and classification models, but despite some past usage, they appear to have been rather neglected as a general-purpose technique.
Abstract: Gaussian processes are a natural way of specifying prior distributions over functions of one or more input variables. When such a function defines the mean response in a regression model with Gaussian errors, inference can be done using matrix computations, which are feasible for datasets of up to about a thousand cases. The covariance function of the Gaussian process can be given a hierarchical prior, which allows the model to discover high-level properties of the data, such as which inputs are relevant to predicting the response. Inference for these covariance hyperparameters can be done using Markov chain sampling. Classification models can be defined using Gaussian processes for underlying latent values, which can also be sampled within the Markov chain. Gaussian processes are in my view the simplest and most obvious way of defining flexible Bayesian regression and classification models, but despite some past usage, they appear to have been rather neglected as a general-purpose technique. This may be partly due to a confusion between the properties of the function being modeled and the properties of the best predictor for this unknown function.
TL;DR: Application of the proposed approach indicates that proposed method effectively captures the nonlinear dynamics in the process variables in the Tennessee Eastman process.
27 Jul 2009
TL;DR: A Monte-Carlo kd-tree sampling algorithm that efficiently computes any filter that can be expressed in this way, along with a GPU implementation of this technique and a fast adaptation of non-local means to geometry.
Abstract: We propose a method for accelerating a broad class of non-linear filters that includes the bilateral, non-local means, and other related filters. These filters can all be expressed in a similar way: First, assign each value to be filtered a position in some vector space. Then, replace every value with a weighted linear combination of all values, with weights determined by a Gaussian function of distance between the positions. If the values are pixel colors and the positions are (x, y) coordinates, this describes a Gaussian blur. If the positions are instead (x, y, r, g, b) coordinates in a five-dimensional space-color volume, this describes a bilateral filter. If we instead set the positions to local patches of color around the associated pixel, this describes non-local means. We describe a Monte-Carlo kd-tree sampling algorithm that efficiently computes any filter that can be expressed in this way, along with a GPU implementation of this technique. We use this algorithm to implement an accelerated bilateral filter that respects full 3D color distance; accelerated non-local means on single images, volumes, and unaligned bursts of images for denoising; and a fast adaptation of non-local means to geometry. If we have n values to filter, and each is assigned a position in a d-dimensional space, then our space complexity is O(dn) and our time complexity is O(dn log n), whereas existing methods are typically either exponential in d or quadratic in n.
TL;DR: A general framework for combining regularized regression methods with the estimation of Graphical Gaussian models is investigated, which includes various existing methods as well as two new approaches based on ridge regression and adaptive lasso, respectively.
Abstract: Graphical Gaussian models are popular tools for the estimation of (undirected) gene association networks from microarray data. A key issue when the number of variables greatly exceeds the number of samples is the estimation of the matrix of partial correlations. Since the (Moore-Penrose) inverse of the sample covariance matrix leads to poor estimates in this scenario, standard methods are inappropriate and adequate regularization techniques are needed. Popular approaches include biased estimates of the covariance matrix and high-dimensional regression schemes, such as the Lasso and Partial Least Squares. In this article, we investigate a general framework for combining regularized regression methods with the estimation of Graphical Gaussian models. This framework includes various existing methods as well as two new approaches based on ridge regression and adaptive lasso, respectively. These methods are extensively compared both qualitatively and quantitatively within a simulation study and through an application to six diverse real data sets. In addition, all proposed algorithms are implemented in the R package "parcor", available from the R repository CRAN. In our simulation studies, the investigated non-sparse regression methods, i.e. Ridge Regression and Partial Least Squares, exhibit rather conservative behavior when combined with (local) false discovery rate multiple testing in order to decide whether or not an edge is present in the network. For networks with higher densities, the difference in performance of the methods decreases. For sparse networks, we confirm the Lasso's well known tendency towards selecting too many edges, whereas the two-stage adaptive Lasso is an interesting alternative that provides sparser solutions. In our simulations, both sparse and non-sparse methods are able to reconstruct networks with cluster structures. On six real data sets, we also clearly distinguish the results obtained using the non-sparse methods and those obtained using the sparse methods where specification of the regularization parameter automatically means model selection. In five out of six data sets, Partial Least Squares selects very dense networks. Furthermore, for data that violate the assumption of uncorrelated observations (due to replications), the Lasso and the adaptive Lasso yield very complex structures, indicating that they might not be suited under these conditions. The shrinkage approach is more stable than the regression based approaches when using subsampling.
TL;DR: It is shown that both the uniform and Gaussian sensor node deployments behave qualitatively in a similar way with respect to the beamwidths and sidelobe levels, while the Gaussian deployment gives wider mainlobe and has lower chance of large sidelobes.
Abstract: Collaborative beamforming has been recently introduced in the context of wireless sensor networks (WSNs) to increase the transmission range of individual sensor nodes. The challenge in using collaborative beamforming in WSNs is the uncertainty regarding the sensor node locations. However, the actual sensor node spatial distribution can be modeled by a properly selected probability density function (pdf). In this paper, we model the spatial distribution of sensor nodes in a cluster of WSN using Gaussian pdf. Gaussian pdf is more suitable in many WSN applications than, for example, uniform pdf which is commonly used for flat ad hoc networks. The average beampattern and its characteristics, the distribution of the beampattern level in the sidelobe region, and the distribution of the maximum sidelobe peak are derived using the theory of random arrays. We show that both the uniform and Gaussian sensor node deployments behave qualitatively in a similar way with respect to the beamwidths and sidelobe levels, while the Gaussian deployment gives wider mainlobe and has lower chance of large sidelobes.
TL;DR: In this paper, the authors study the convergence power spectrum and its covariance for a standard ΛCDM cosmology and compare the simulation results with analytic model predictions, and argue that a prior knowledge on the full distribution may be needed to obtain an unbiased estimate on the ensemble-averaged band power at each angular scale from a finite volume survey.
Abstract: We study the lensing convergence power spectrum and its covariance for a standard ΛCDM cosmology. We run 400 cosmological N-body simulations and use the outputs to perform a total of 1000 independent ray-tracing simulations. We compare the simulation results with analytic model predictions. The semianalytic model based on Smith et al. fitting formula underestimates the convergence power by ~ 30% at arcmin angular scales. For the convergence power spectrum covariance, the halo model reproduces the simulation results remarkably well over a wide range of angular scales and source redshifts. The dominant contribution at small angular scales comes from the sample variance due to the number fluctuations of halos in a finite survey volume. The signal-to-noise ratio for the convergence power spectrum is degraded by the non-Gaussian covariances by up to a factor of 5 for a weak lensing survey to zs ~ 1. The probability distribution of the convergence power spectrum estimators, among the realizations, is well approximated by a χ2 distribution with broadened variance given by the non-Gaussian covariance, but has a larger positive tail. The skewness and kurtosis have non-negligible values especially for a shallow survey. We argue that a prior knowledge on the full distribution may be needed to obtain an unbiased estimate on the ensemble-averaged band power at each angular scale from a finite volume survey.
TL;DR: Second-order Poincare inequalities (SOPE inequalities) as discussed by the authors were introduced to derive gaussian central limit theorems for Gaussian Toeplitz matrices.
Abstract: Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems The proofs of such results are usually rather difficult, involving hard computations specific to the model in question In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments In the process, we introduce a notion of ‘second order Poincare inequalities’: just as ordinary Poincare inequalities give variance bounds, second order Poincare inequalities give central limit theorems The proof of the main result employs Stein’s method of normal approximation A number of examples are worked out, some of which are new One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices
TL;DR: A simple and consistent informational framework of the common patterns of nature based on the method of maximum entropy shows that each neutral generative model is a special case that helps to discover a particular set of informational constraints; those informational constraints define a much wider domain of non‐neutral generative processes that attract to the same neutral pattern.
Abstract: We typically observe large-scale outcomes that arise from the interactions of many hidden, small-scale processes. Examples include age of disease onset, rates of amino acid substitutions and composition of ecological communities. The macroscopic patterns in each problem often vary around a characteristic shape that can be generated by neutral processes. A neutral generative model assumes that each microscopic process follows unbiased or random stochastic fluctuations: random connections of network nodes; amino acid substitutions with no effect on fitness; species that arise or disappear from communities randomly. These neutral generative models often match common patterns of nature. In this paper, I present the theoretical background by which we can understand why these neutral generative models are so successful. I show where the classic patterns come from, such as the Poisson pattern, the normal or Gaussian pattern and many others. Each classic pattern was often discovered by a simple neutral generative model. The neutral patterns share a special characteristic: they describe the patterns of nature that follow from simple constraints on information. For example, any aggregation of processes that preserves information only about the mean and variance attracts to the Gaussian pattern; any aggregation that preserves information only about the mean attracts to the exponential pattern; any aggregation that preserves information only about the geometric mean attracts to the power law pattern. I present a simple and consistent informational framework of the common patterns of nature based on the method of maximum entropy. This framework shows that each neutral generative model is a special case that helps to discover a particular set of informational constraints; those informational constraints define a much wider domain of non-neutral generative processes that attract to the same neutral pattern.
TL;DR: In this paper, a special mean-field problem in a purely stochastic approach is investigated for the solution (Y, Z) of a mean field backward stochastastic differential equation with solution X, where coefficients are governed by N independent copies of (X-N, Y, N, Z, Z(N)).
Abstract: Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean-Vlasov type with solution X we study a special approximation by the solution (X-N, Y-N, Z(N)) of some decoupled forward-backward equation which coefficients are governed by N independent copies of (X-N, Y-N, Z(N)). We show that the convergence speed of this approximation is of order 1/root N. Moreover, our special choice of the approximation allows to characterize the limit behavior of root N(X-N - X, Y-N - Y, Z(N) - Z). We prove that this triplet converges in law to the solution of some forward-backward. stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.
TL;DR: In this article, the authors give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension is large compared to the sample size.
Abstract: In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension is large compared to the sample size. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F-matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with chi-square approximation fails. Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.