Nonnegative Kernel Research Articles

Stability with respect to initial conditions in V-norm for nonlinear filters with ergodic observations

AbstractWe establish conditions for an exponential rate of forgetting of the initial distribution of nonlinear filters in V-norm, allowing for unbounded test functions. The analysis is conducted in an general setup involving nonnegative kernels in a random environment which allows treatment of filters and prediction filters in a single framework. The main result is illustrated on two examples, the first showing that a total variation norm stability result obtained by Douc et al. (2009) can be extended to V-norm without any additional assumptions, the second concerning a situation in which forgetting of the initial condition holds in V-norm for the filters, but the V-norm of each prediction filter is infinite.

Construction of weighted crystallographic orientations capturing a given orientation density function

To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallee Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about \(99.75\,\%\) to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than \(10\,\%\).

On the strong Brillinger-mixing property of α-determinantal point processes and some applications

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function C(x, y) defining an α-determinantal point process (DPP). Assuming absolute integrability of the function C 0(x) = C(o, x), we show that a stationary α-DPP with kernel function C 0(x) is “strongly” Brillinger-mixing, implying, among others, that its tail-σ-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications to statistical second-order analysis of α-DPPs.

Decay bounds for nonlocal evolution equations in Orlicz spaces

We show decay bounds of the form ∫Rdϕ(u(x,t))dx≤Ct−μ for integrable and bounded solutions to the nonlocal evolution equation ut(x,t)=∫RdJ(x,y)G(u(y,t)−u(x,t))(u(y,t)−u(x,t))dy+f(u(x,t)). Here G is a nonnegative and even function, and f verifies f(ξ)ξ≤0 for all ξ≥0. We remark that G is not assumed to be homogeneous. The function ϕ and the exponent μ depend on G via adequate hypotheses, while J is a nonnegative kernel satisfying suitable assumptions.

Sharp constants for a class of multilinear integral operators and some applications

We investigate a class of multilinear integral operators with the nonnegative kernels, and prove that the norms of the operators can be obtained by integral of the product of the kernel function and finitely many basic functions. Using the integral, we can easily calculate the sharp constants for the multilinear Hilbert inequality, the generalized Hardy-Littlewood-Sobolev inequality and the multilinear Hardy operator.

One-dimensional solutions of non-local Allen–Cahn-type equations with rough kernels

We are interested in the study of local and global minimizers for an energy functional of the type14∬R2N∖(RN∖Ω)2|u(x)−u(y)|2K(x−y)dxdy+∫ΩW(u(x))dx, where W is a smooth, even double-well potential and K is a non-negative symmetric kernel in a general class, which contains as a particular case the choice K(z)=|z|−N−2s, with s∈(0,1), related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in the full space RN and obtain sharp estimates for some quantities associated to it. In particular, we deduce the existence of solutions of the non-local Allen–Cahn equationp.v.∫RN(u(x)−u(y))K(x−y)dy+W′(u(x))=0for any x∈RN, which possess one-dimensional symmetry.The results presented here were proved in [9,10,36] for the model case K(z)=|z|−N−2s. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity.

Dynamics and invariant manifolds for a nonlocal stochastic Swift-Hohenberg equation

The dynamics and invariant manifolds for a nonlocal stochastic Swift-Hohenberg equation with multiplicative noise are investigated. Using a stochastic transformation process, a nonlocal stochastic Swift-Hohenberg equation is studied with either a positive kernel or a non-negative kernel. Then the dynamics, existence, and uniqueness of a global random attractor for the nonlocal stochastic Swift-Hohenberg equation is shown. Moreover, the existence of a local random invariant manifold of the corresponding random dynamical system for the nonlocal stochastic Swift-Hohenberg equation with multiplicative noise is obtained using the technique of a cut-off function and the Lyapunov-Perron method.

A nonlocal dispersal logistic equation with spatial degeneracy

In this paper, we study the nonlocal dispersal Logistic equation \begin{equation*} \begin{cases} u_t=Du+\lambda m(x)u-c(x)u^p &\text{ in }{\Omega}\times(0,+\infty),\\ u(x,0)=u_0(x)\geq0&\text{ in }{\Omega}, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain, $\lambda>0$ and $p>1$ are constants. $Du(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy$ represents the nonlocal dispersal operator with continuous and nonnegative dispersal kernel $J$, $m\in C(\bar{\Omega})$ and may change sign in $\Omega$. The function $c$ is nonnegative and has a degeneracy in some subdomain of $\Omega$. We establish the existence and uniqueness of positive stationary solution and also consider the effect of degeneracy of $c$ on the long-time behavior of positive solutions. Our results reveal that the necessary condition to guarantee a positive stationary solution and the asymptotic behaviour of solutions are quite different from those of the corresponding reaction-diffusion equation.

Face Recognition Based on Wavelet Kernel Non-Negative Matrix Factorization

Abstract In this paper a novel face recognition algorithm, based on wavelet kernel non-negative matrix factorization (WKNMF), is proposed. By utilizing features from multi-resolution analysis, the nonlinear mapping capability of kernel nonnegative matrix factorization could be improved by the method proposed. The proposed face recognition method combines wavelet kernel non-negative matrix factorization and RBF network. Extensive experimental results on ORL and YALE face database show that the suggested method possesses much stronger analysis capability than the comparative methods. Compared with PCA, non-negative matrix factorization, kernel PCA and independent component analysis, the proposed face recognition method with WKNMF and RBF achieves over 10 % improvement in recognition accuracy.

Some new extensions of Hardy's inequality

In this study, by a non-negative homogeneous kernel k we prove some extensions of Hardy's inequalityin two and three dimensions

TRAVELING WAVE SOLUTIONS OF NONLINEAR SCALAR INTEGRAL DIFFERENTIAL EQUATIONS ARISING FROM SYNAPTICALLY COUPLED NEURONAL NETWORKS

Consider the following nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks. These model equations generalize many important nonlinear scalar integral differential equations aris ing from synaptically coupled neuronal networks. The kernel functions K and W represent synaptic couplings between neurons in synaptically coupled neuronal networks. The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions), lateral inhibitions (modeled with Mexican hat kernel functions), lateral excitations (modeled with upside down Mexican hat kernel functions), but also synaptic couplings which may change sign for finitely many times or even infinitely many times. The function H = H(u − ) represents the Heaviside step function, which is defined by H(u − ) = 0 for all u . The functions and represent probability density functions defined on (0,1). The parameter c > 0 represents the speed of an action potential and the parameter > 0 represents a constant delay. In these equations, u = u(x, t) stands for the membrane potential of a neuron at position x and time t. The positive constants > 0 and > 0 represent synaptic rates. The positive constants > 0 and > 0 represent thresholds for excitation of neurons. The function f = f(u) represents the sodium currents in neuronal networks. The positive constant w 0 > 0 is to be given. The authors will establish the existence and stability of traveling wave solutions of these nonlinear scalar integral differential equations by coupling together speed index functions, stability index func tions (often called Evans functions, that is, complex analytic functions), implicit function theorem, intermediate value theorem, mean value theorem, global strong maximum principle for Evans func tions, linearized stability criterion and many other important techniques in dynamical systems. They will find sufficient conditions satisfied by the synaptic couplings, by the probability density functions, by the synaptic rate constants and by the thresholds so that the traveling wave solutions and their wave speeds exist, and the stability of the traveling wave solutions is true. The main results obtained in this paper greatly improve many previous results.

Multivariate Transient Price Impact and Matrix-Valued Positive Definite Functions

We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such strategies is guaranteed by assuming that the decay kernel corresponds to a matrix-valued positive definite function. An example illustrates, however, that positive definiteness alone does not guarantee that optimal strategies are well-behaved. Building on previous results from the one-dimensional case, we investigate a class of nonincreasing, non-negative, and convex decay kernels with values in a space of symmetric matrices. We show that these decay kernels are always positive definite and characterize when they are even strictly positive definite, a result that may be of independent interest. Optimal strategies for kernels from this class are particularly well-behaved if one requires that the decay kernel is also commuting. We show how such decay kernels can be constructed by means of matrix functions and provide a number of examples. In particular, we completely solve the case of matrix exponential decay.

Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks

Consider the following nonlinear scalar integral differential equation arising from delayed synaptically coupled neuronal networks\begin{eqnarray*}\frac{\partial u}{\partial t}+f(u)&=&\alpha\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y)H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\&+&\beta\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H(u(y,t-\tau)-\Theta){\rm d}y\right]{\rm d}\tau.\end{eqnarray*}This model equation generalizes many important nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks.The kernel functions $K$ and $W$ represent synaptic couplings between neurons in synaptically coupled neuronal networks.The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions),but also lateral inhibitions (modeled with Mexican hat kernel functions) and lateral excitations (modeled with upside down Mexican hat kernel functions).In this nonlinear scalar integral differential equation, $u=u(x,t)$ stands for the membrane potential of a neuron at position $x$ and time $t$.The integrals represent nonlocal spatio-temporal interactions between neurons.   &nbsp We have accomplished the existence and stability of three traveling wave fronts $u(x,t)=U_k(x+\mu_kt)$ of the nonlinear scalar integral differential equation in an earlier work [42],where $\mu_k$ denotes the wave speed and $z=x+\mu_kt$ denotes the moving coordinate, $k=1,2,3$.In this paper, we will investigate how the neurobiological mechanisms represented by the synaptic couplings $(K,W)$,by the probability density functions $(\xi,\eta)$, by the synaptic rate constants $(\alpha,\beta)$ and by the firingthresholds $(\theta,\Theta)$ influence the wave speeds $\mu_k$ of the traveling wave fronts.We will define several speed index functions and use rigorous mathematical analysis to investigate the influence of the neurobiological mechanisms on the wave speeds.In particular, we will compare wave speeds of the traveling wave fronts of the nonlinear scalar integral differential equationwith different synaptic couplings and with different probability density functions;we will accomplish new asymptotic behaviors of the wave speeds; we will compare wave speeds of traveling wave fronts ofmany reduced forms of nonlinear scalar integral differential equations of the above model equation;we will establish new estimates of the wave speeds. All these will greatly improve results obtained in previous work [38], [40] and [41].

Approximating Markov chains and [formula omitted]-geometric ergodicity via weak perturbation theory

Let P be a Markov kernel on a measurable space X and let V:X→[1,+∞). This paper provides explicit connections between the V-geometric ergodicity of P and that of finite-rank non-negative sub-Markov kernels P̂k approximating P. A special attention is paid to obtain an efficient way to specify the convergence rate for P from that of P̂k and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the P-invariant probability measure and the P̂k-invariant positive measure. The proofs are based on the Keller–Liverani perturbation theorem which requires an accurate control of the essential spectral radius of P on usual weighted supremum spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of P is applied to truncation of discrete Markov kernels on X:=N.

On improving convergence rate of Bernstein polynomial density estimator

This paper is concerned with the Bernstein estimator [Vitale, R.A. (1975), ‘A Bernstein Polynomial Approach to Density Function Estimation’, in Statistical Inference and Related Topics, ed. M.L. Puri, 2, New York: Academic Press, pp. 87–99] to estimate a density with support [0, 1]. One of the major contributions of this paper is an application of a multiplicative bias correction [Terrell, G.R., and Scott, D.W. (1980), ‘On Improving Convergence Rates for Nonnegative Kernel Density Estimators’, The Annals of Statistics, 8, 1160–1163], which was originally developed for the standard kernel estimator. Moreover, the renormalised multiplicative bias corrected Bernstein estimator is studied rigorously. The mean squared error (MSE) in the interior and mean integrated squared error of the resulting bias corrected Bernstein estimators as well as the additive bias corrected Bernstein estimator [Leblanc, A. (2010), ‘A Bias-reduced Approach to Density Estimation Using Bernstein Polynomials’, Journal of Nonparametric Statistics, 22, 459–475] are shown to be O(n−8/9) when the underlying density has a fourth-order derivative, where n is the sample size. The condition under which the MSE near the boundary is O(n−8/9) is also discussed. Finally, numerical studies based on both simulated and real data sets are presented.

Spatiotemporal mutualistic model of mistletoes and birds

A mathematical model which incorporates the spatial dispersal and interaction dynamics of mistletoes and birds is derived and studied to gain insights of the spatial heterogeneity in abundance of mistletoes. Fickian diffusion and chemotaxis are used to model the random movement of birds and the aggregation of birds due to the attraction of mistletoes, respectively. The spread of mistletoes by birds is expressed by a dispersal operator, which is typically a convolution integral with a dispersal kernel. Two different types of kernel functions are used to study the model, one is a Dirac delta function which reflects the special case that the spread behavior is local, and the other one is a general non-negative symmetric function which describes the nonlocal spread of mistletoes. When the kernel function is taken as the Dirac delta function, the threshold condition for the existence of mistletoes is given and explored in terms of parameters. For the general non-negative symmetric kernel case, we prove the existence and stability of spatially nonhomogeneous equilibria. Numerical simulations are conducted by taking specific forms of kernel functions. Our study shows that the spatial heterogeneous patterns of mistletoes are related to the specific dispersal pattern of birds which carry mistletoe seeds.

Kernel density estimation on the rotation group and its application to crystallographic texture analysis

We are concerned with kernel density estimation on the rotation group SO(3). We prove asymptotically optimal convergence rates for the minimax risk of the mean integrated squared error for different function classes including bandlimited functions, functions with bounded Sobolev norm and functions with polynomially decaying Fourier coefficients and give optimal kernel functions. Furthermore, we consider kernel density estimation with nonnegative kernel functions and prove analogous saturation behavior as it is known for the Euclidean case, i.e., the optimal minimax rate does not improve for smoothness classes of functions which are more than two times differentiable. We also benchmark several families of kernel functions with respect to their capability for kernel density estimation. To make our finding applicable, we give a fast algorithm for the computation of the kernel density estimator for large sampling sets and illustrate our theoretical findings by numerical experiments. Finally, we apply our results to answer a long standing question in crystallographic texture analysis on the number of orientation measurements needed to estimate the underlying orientation density function up to a given accuracy.

Singular measure as principal eigenfunction of some nonlocal operators

In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution (λ,ϕ) of a nonlocal operator: ∫ΩK(x,y)ϕ(y)dy+a(x)ϕ(x)=−λϕ(x), where Ω⊂Rn is a bounded domain, K is a nonnegative kernel and a is continuous. We prove that for the generalised principal eigenvalue λp≔sup{λ∈R∣∃ϕ∈C(Ω),ϕ>0 so that LΩ[ϕ]+a(x)ϕ+λϕ≤0} there exists always a solution (dμ,λp) of the problem in the space of positive measure. When dμ is absolutely continuous with respect to the Lebesgue measure, dμ=ϕp(x)dx is called the principal eigenfunction associated with λp. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem.

A nonparametric method for estimating asymmetric densities based on skewed Birnbaum–Saunders distributions applied to environmental data

In this paper, we introduce a new nonparametric kernel method for estimating asymmetric densities based on generalized skew-Birnbaum–Saunders distributions. Kernels based on these distributions have the advantage of providing flexibility in the asymmetry and kurtosis levels. In addition, the generalized skew-Birnbaum–Saunders kernel density estimators are boundary bias free and achieve the optimal rate of convergence for the mean integrated squared error of the nonnegative asymmetric kernel estimators. We carry out a data analysis consisting of two parts. First, we conduct a Monte Carlo simulation study for evaluating the performance of the proposed method. Second, we use this method for estimating the density of three real air pollutant concentration data sets. These numerical results favor the proposed nonparametric estimators.

Traveling waves of a non-local conservation law

We consider traveling waves related to shocks for a non-local scalar conservation law \[ u_{t}+(f(u))_{x}=K\ast u-u, \] where $f$ is an arbitrary convex function, $K\ast u$ stands for the convolution in the spatial variable $x$, and $K$ is an arbitrary non-negative kernel with unit mass centered at the origin. Given a pair of values at $\pm\infty$ with $u_{-} > u_{+}$, we establish here the existence and uniqueness of a traveling-wave solution $u=\phi\left( x-ct\right) $ such that the wave speed \[ c=\frac{f\left( u_{+}\right) -f\left( u_{-}\right) }{u_{+}-u_{-}} \] and $\phi$, satisfying $\phi\left( \pm\infty\right) =u_{\pm}$, is a smooth decreasing function that can have at most one jump discontinuity at the origin. Our approach here is to consider a family of ``truncated" problems on $\left[ -n,n\right] $ which can be solved using Schauder's fixed-point theorem and then sending $n\rightarrow\infty$.