Class Of Kernels Research Articles

Fast Exact Evaluation of Univariate Kernel Sums.

This paper presents new methodology for computationally efficient evaluation of univariate kernel sums. It is shown that a rich class of kernels allows for exact evaluation of functions expressed as a sum of kernels using simple recursions. Given an ordered sample the computational complexity is linear in the sample size. Direct applications to the estimation of denisties and their derivatives shows that the proposed approach is competitive with the state-of-the-art. Extensions to multivariate problems including independent component analysis and spatial smoothing illustrate the versatility of univariate kernel estimators, and highlight the efficiency and accuracy of the proposed approach. Multiple applications in image processing, including image deconvolution; denoising; and reconstruction are considered, showing that the proposed approach offers very promising potential in these fields.

Locally Homogeneous Covariance Matrix Representation for Hyperspectral Image Classification

Combining spectralandspatial information has been proven to be an effective way for hyperspectral image (HSI) classification. However, making full use of spectral–spatial information of HSI still remains an open problem, especially when only a small number of labeled samples are available. In this article, a new spectral–spatial feature extraction method called locally homogeneous covariance matrix representation (CMR) is proposed for the fusion of spectral and spatial information. Specially, to make use of neighborhood homogeneity of land covers, original HSI is first segmented into many superpixels using modified entropy rate superpixel segmentation. Then, to acquire the most similar pixels, we propose to construct neighborhoods of each pixel from the overlapping areas between the corresponding superpixels and the sliding window centered on it. Subsequently, CMRs of different pixels can be obtained. In the classification stage, we fed the obtained CMRs into SVM with Log-Euclidean-based kernel for classification. Compared to the traditional approach that utilizes neighboring information only within a fixed window, the proposed local homogeneity strategy can absorb more discriminative spectral–spatial features. Experimental results from a series of available HSI datasets show that our proposed method is superior to several state-of-the-art methods, especially when the training set is very limited.

An Energy Stable and Maximum Bound Preserving Scheme with Variable Time Steps for Time Fractional Allen--Cahn Equation

In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order $\alpha \rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $\alpha \rightarrow 1$ limit. This seems to be the first work on variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is build upon a reformulated problem associated with the Riemann-Liouville derivative. As a by product, we build up a reversible transformation between the L1-type formula of the Riemann-Liouville derivative and a new L1-type formula of the Caputo derivative, with the help of a class of discrete orthogonal convolution kernels. This is the first time such a \textit{discrete} transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed scheme.

Characterization of compact linear integral operators in the space of functions of bounded variation

Although various operators in the space of functions of bounded variation have been studied by quite a few authors, no simple necessary and sufficient conditions guaranteeing compactness of linear integral operators acting in such spaces have been known. The aim of the paper is to fully characterize the class of kernels which generate compact linear integral operators in the BV-space. Using this characterization we show that certain weakly singular and convolution operators (such as the Abel and Volterra operators), when considered as transformations of \(BV[a,b]\), are compact. We also provide a detailed comparison of those new necessary and sufficient conditions with various other conditions connected with compactness of linear (integral) operators in the space of functions of bounded variation which already exist in the literature.

Logistic Regression Based on Statistical Learning Model with Linearized Kernel for Classification

In this paper, we propose a logistic regression classification method based on the integration of a statistical learning model with linearized kernel pre-processing. The single Gaussian kernel and fusion of Gaussian and cosine kernels are adopted for linearized kernel pre-processing respectively. The adopted statistical learning models are the generalized linear model and the generalized additive model. Using a generalized linear model, the elastic net regularization is adopted to explore the grouping effect of the linearized kernel feature space. Using a generalized additive model, an overlap group-lasso penalty is used to fit the sparse generalized additive functions within the linearized kernel feature space. Experiment results on the Extended Yale-B face database and AR face database demonstrate the effectiveness of the proposed method. The improved solution is also efficiently obtained using our method on the classification of spectra data.

Hardware Acceleration of High Sensitivity Power-Aware Epileptic Seizure Detection System Using Dynamic Partial Reconfiguration

In this paper, a high-sensitivity low-cost power-aware Support Vector Machine (SVM) training and classification based system, is hardware implemented for a neural seizure detection application. The training accelerator algorithm, adopted in this work, is the sequential minimal optimization (SMO). System blocks are implemented to achieve the best trade-off between sensitivity and the consumption of area and power. The proposed seizure detection system achieves 98.38% sensitivity when tested with the implemented linear kernel classifier. The system is implemented on different platforms: such as Field Programmable Gate Array (FPGA) Xilinx Virtex-7 board and Application Specific Integrated Circuit (ASIC) using hardware-calibrated UMC 65nm CMOS technology. A power consumption evaluation is performed on both the ASIC and FPGA platforms showing that the ASIC power consumption is lower by at least 65% when compared with the FPGA counterpart. A power-aware system is implemented with FPGAs by the adoption of the Dynamic Partial Reconfiguration (DPR) technique that allows the dynamic operation of the system based on power level available to the system at the expense of degradation of the system accuracy. The proposed system exploits the advantages of DPR technology in FPGAs to switch between two proposed designs providing a decrease of 64% in power consumption.

A New Convolutional Kernel Classifier for Hyperspectral Image Classification

Multiple Kernel Learning (MKL) algorithms are among the most successful classification methods for hyperspectral data. Nevertheless, these algorithms suffer from two main drawbacks of computational complexity and debility to admit to the end-to-end learning paradigm. This paper proposed a Convolutional Kernel Classifier (CKC) for hyperspectral remote sensing images to address these issues. The CKC uses the Nystrm approximation method to estimate a low-rank approximation of the basis kernels, thus solves the issues associated with the high dimensionality of the basis kernels. The CKC uses deep architecture to learn the optimal combination of the basis kernels and the classification task to enable end-to-end learning. The proposed CKCs architecture is based on a 1D-Convolutional Neural Network (CNN-1D), and it uses kernel dropout to prevent overfitting. It is the first instance of deep-kernel algorithms in the field of remote sensing. The proposed method was compared with several well-known hyperspectral image analysis MKL algorithms, including a multi-kernel variant of the deep kernel machine optimization (M-DKMO), MKL-average, Simple-MKL, and Generalize MKL (GMKL), and state-of-the-art deep learning models, including Vanilla Recurrent Neural Network (VanillaRNN) and CNN-1D in classifying four benchmark hyperspectral datasets. The experimental results show that the CKC consistently outperforms all the competitor methods, and its runtime is lower than its MKL algorithm counterparts on four benchmark hyperspectral datasets. Moreover, the Nystrm approximation solves the high dimensionality of the basis kernels and boosts classification accuracy. The source codes of CKC are available from: https://github.com/MohsenAnsari1373/A-New-Convolutional-Kernel-Classifier-for-Hyperspectral-Image-Classification .

Variable bandwidth kernel regression estimation

In this paper we propose a variable bandwidth kernel regression estimator fori.i.d. observations in ℝ2to improve the classical Nadaraya-Watson estimator. The bias is improved to the order ofO(hn4) under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.

Retracted] Remote Judgment Method of Painting Image Style Plagiarism Based on Wireless Network Multitask Learning

Since the artistry of the work cannot be accurately described, the identification of reproducible plagiarism is more difficult. The identification of reproducible plagiarism of digital image works requires in‐depth research on the artistry of artistic works. In this paper, a remote judgment method for plagiarism of painting image style based on wireless network multitask learning is proposed. According to this new method, the uncertainty of painting image samples is removed based on multitask learning algorithm edge sampling. The deep‐level details of the painting image are extracted through the multitask classification kernel function, and most of the pixels in the image are eliminated. When the clustering density is greater than the judgment threshold, it can be considered that the two images have spatial consistency. It can also be judged based on this that the two images are similar, that is, there is plagiarism in the painting. The experimental results show that the discrimination rate is always close to 100%, the misjudgment rate of plagiarism of painting images has been reduced, and the various indicators in the discrimination process are the lowest, which fully shows that a very satisfactory discrimination result can be obtained.

Optimal design for kernel interpolation: Applications to uncertainty quantification

The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation using radial basis functions (RBFs) in a stable way, we propose a type of quasi-optimal interpolation points, searching from a large set of candidate points, using a procedure similar to designing Fekete points or power function maximizing points that use pivot from a Cholesky decomposition. The proposed quasi-optimal points results in smaller condition number, and thus mitigates the instability of the interpolation procedure when the number of points becomes large. Applications to parametric uncertainty quantification are presented, and it is shown that the proposed interpolation method can outperform sparse grid methods in many interesting cases. We also demonstrate the new procedure can be applied to constructing gradient-enhanced Gaussian process emulators.

Application of an ensemble earthquake rate model in Italy, considering seismic catalogs and fault moment release

We develop an ensemble earthquake rate model that provides spatially variable time-independent (Poisson) long-term annual occurrence rates of seismic events throughout Italy, for magnitude bin of 0.1 units from Mw ≥ 4.5 in spatial cells of 0.1° × 0.1°. We weighed seismic activity rates of smoothed seismicity and fault-based inputs to build our earthquake rupture forecast model, merging it into a single ensemble model. Both inputs adopt a tapered Gutenberg-Richter relation with a single b-value and a single corner magnitude estimated by earthquakes catalog. The spatial smoothed seismicity was obtained using the classical kernel smoothing method with the inclusion of magnitude dependent completeness periods applied to the Historical (CPTI15) and Instrumental seismic catalogs. For each seismogenic source provided by the Database of the Individual Seismogenic Sources (DISS), we computed the annual rate of the events above Mw 4.5, assuming that the seismic moments of the earthquakes generated by each fault are distributed according to the tapered Gutenberg-Richter relation with the same parameters of the smoothed seismicity models. Comparing seismic annual rates of the catalogs with those of the seismogenic sources, we realized that there is a good agreement between these rates in Central Apennines zones, whereas the seismogenic rates are higher than those of the catalogs in the north east and south of Italy. We also tested our model against the strong Italian earthquakes (Mw 5.5+), in order to check if the total number (N-test) and the spatial distribution (S-test) of these events was compatible with our model, obtaining good results, i.e. high p-values in the test. The final model will be a branch of the new Italian seismic hazard map.

Analysis of adaptive BDF2 scheme for diffusion equations

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k:=\tau_k/\tau_{k-1}\le(3+\sqrt{17})/2\approx3.561$, the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the $L^2$ norm. The second-order temporal convergence can be recovered if almost all of time-step ratios $r_k\le 1+\sqrt{2}$ or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the $H^1$ seminorm) and the $L^2$ norm monotonicity at the discrete levels. An example is included to support our analysis.

Breakdown of Liesegang precipitation bands in a simplified fast reaction limit of the Keller\u2013Rubinow model

We study solutions to the integral equation ω(x)=Γ-x2∫01K(θ)H(ω(xθ))dθ\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\omega (x) = \\Gamma - x^2 \\int \\nolimits _{0}^1 K(\\theta ) \\, H(\\omega (x\\theta )) \\, \\mathrm {d}\\theta \\end{aligned}$$\\end{document}where Gamma >0, K is a weakly degenerate kernel satisfying, among other properties, K(theta ) sim k , (1-theta )^sigma as theta rightarrow 1 for constants k>0 and sigma in (0, log _2 3 -1), H denotes the Heaviside function, and x in [0,infty ). This equation arises from a reaction-diffusion equation describing Liesegang precipitation band patterns under certain simplifying assumptions. We argue that the integral equation is an analytically tractable paradigm for the clustering of precipitation rings observed in the full model. This problem is nontrivial as the right hand side fails a Lipschitz condition so that classical contraction mapping arguments do not apply. Our results are the following. Solutions to the integral equation, which initially feature a sequence of relatively open intervals on which omega is positive (“rings”) or negative (“gaps”) break down beyond a finite interval [0,x^*] in one of two possible ways. Either the sequence of rings accumulates at x^* (“non-degenerate breakdown”) or the solution cannot be continued past one of its zeroes at all (“degenerate breakdown”). Moreover, we show that degenerate breakdown is possible within the class of kernels considered. Finally, we prove existence of generalized solutions which extend the integral equation past the point of breakdown.

Mellin–Meijer kernel density estimation on $${{\mathbb {R}}}^+$$

Kernel density estimation is a nonparametric procedure making use of the smoothing power of the convolution operation. Yet, it performs poorly when the density of a positive variable is estimated, due to boundary issues. So, various extensions of the kernel estimator allegedly suitable for $${\mathbb {R}}^+$$ -supported densities, such as those using asymmetric kernels, abound in the literature. Those, however, are not based on any valid smoothing operation. By contrast, in this paper a kernel density estimator is defined through the Mellin convolution, the natural analogue on $${\mathbb {R}}^+$$ of the usual convolution. From there, a class of asymmetric kernels related to Meijer G-functions is suggested, and asymptotic properties of this ‘Mellin–Meijer kernel density estimator’ are presented. In particular, its pointwise- and $$L_2$$ -consistency (with optimal rate of convergence) are established for a large class of densities, including densities unbounded at 0 and showing power-law decay in their right tail.

New general decay result for a system of viscoelastic wave equations with past history

<p style='text-indent:20px;'>This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels <inline-formula><tex-math id="M1">\begin{document}$ g_i : [0, +\infty) \rightarrow (0, +\infty) $\end{document}</tex-math></inline-formula> satisfying <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\qquad\forall\,t\geq0 \quad\mathrm{and\ for\ }i = 1,2, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ H_i $\end{document}</tex-math></inline-formula> are functions satisfying some specific properties. Under this very general assumption on the behavior of <inline-formula><tex-math id="M4">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity, we establish a relation between the decay rate of the solutions and the growth of <inline-formula><tex-math id="M5">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.

Face Recognition from Small Datasets using Kernel Selection of Gabor Features

Recent advances in face recognition are mostly based on deep methods that require large datasets for training. This paper presents a novel method that combines Gabor-features, feature and kernel to achieve comparable performance on smaller datasets.The paper compares different feature methods in this context.The problem tackled in this paper is achieving accurate face recognition with limited computational resources. By “limited computational resources we mean low computational power (i.e. memory, CPU ops) during both system training and evaluation. Noted that we are not competing against deep learning systems in term of accuracy but we provided a middle ground between hand-coded fast feature extraction and learning based deep learning in terms of both speed and accuracy.To achieve this goal, we propose “kernel selection as the main method to reduce the dimensionality of the classification problem faced by the final classifier in the FR system. Kernel is the process of eliminating less important Gabor kernels for classification while keeping the level of accuracy achievable. Kernel differs from traditional feature in measuring the value of complete kernels consisting of several features together. Because of its structured nature, Kernel has the advantage of eliminating the need to evaluate complete Gabor kernels reducing the computational cost of the system compared with traditional feature methods.

The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect

<p style='text-indent:20px;'>In this paper, we analyze the interaction of localized patterns such as traveling wave solutions for reaction-diffusion systems with nonlocal effect in one space dimension. We consider the case that a nonlocal effect is given by the convolution with a suitable integral kernel. At first, we deduce the equation describing the movement of interacting localized patterns in a mathematically rigorous way, assuming that there exists a linearly stable localized solution for general reaction-diffusion systems with nonlocal effect. When the distances between localized patterns are sufficiently large, the motion of localized patterns can be reduced to the equation for the distances between them. Finally, using this equation, we analyze the interaction of front solutions to some nonlocal scalar equation. Under some assumptions, we can show that the front solutions are interacting attractively for a large class of integral kernels.

Convergence Almost Everywhere of Non-convolutional Integral Operators in Lebesgue Spaces

The case of one-dimensional and multidimensional non-convolutional integral operators in Lebesgue spaces is considered in this paper. The convergence in the norm and almost everywhere of non-convolution integral operators in Lebesgue spaces was insufficiently studied. The kernels <img src=image/13420650_01.gif> of non-convolutional integral operators do not need to have a monotone majorant, so the well-known results on the convergence almost everywhere of convolutional averages are not applicable here. The kernels <img src=image/13420650_01.gif> of nonconvolutional integral operators take into account different behaviors at <img src=image/13420650_02.gif> and <img src=image/13420650_03.gif> depending on <img src=image/13420650_04.gif> (which is important in applications) and cover the situation in the particular case of convolutional and non-convolutional integral operators. We are interested in the behavior of function <img src=image/13420650_05.gif> as <img src=image/13420650_06.gif>. Theorems on convergence almost everywhere in the case of one-dimensional and multidimensional nonconvolution integral operators in Lebesgue spaces are proved. The theorems proved are more general ones (including for convolutional integral operators) and cover a wide class of kernels.

A non-parametric density estimate adaptation for population abundance when the shoulder condition is violated

The non-parametric kernel density estimation is used in practice to estimate population abundance using the line transect sampling. In general, the classical kernel estimator of f(0) tends to be underestimated. In this article, a shifted logarithmic transformation of perpendicular distance is proposed for the kernel estimator when the shoulder condition is violated. Mathematically, the proposed estimator is more efficient than the classical kernel estimator. A simulation study is also carried out to compare the performance of the proposed estimators and the classical kernel estimators.

Robust constrained minimum mixture kernel risk-sensitive loss algorithm for adaptive filtering

Recently, the kernel based constrained adaptive filtering algorithm has attracted a lot of attentions because of its robustness and superiority over traditional methods. As two classic kernel based algorithms, both constrained maximum correntropy criterion (CMCC) and constrained minimum error entropy (CMEE) have shown their superiorities in the case of non-Gaussian noise. However, both algorithms use only one kernel as the kernel function. To further improve the performance of the kernel based adaptive filtering algorithms, we first define the mixture kernel risk-sensitive loss (MKRSL) and study its properties. Then, we apply it to the constrained adaptive filtering and propose a novel constrained minimum MKRSL (CMM-KRSL) algorithm in this paper. Furthermore, we present the performance analysis of the CMM-KRSL algorithm, and provide the stability condition and the theoretical mean square deviation (MSD). Finally, we validate the accuracy of performance analysis and the superiorities of CMM-KRSL by simulations.