Nonnegative Kernel Research Articles

Convergence Analysis of Mean Shift.

The mean shift (MS) algorithm seeks a mode of the kernel density estimate (KDE). This study presents a convergence guarantee of the mode estimate sequence generated by the MS algorithm and an evaluation of the convergence rate, under fairly mild conditions, with the help of the argument concerning the Łojasiewicz inequality. Our findings extend existing ones covering analytic kernels and the Epanechnikov kernel. Those are significant in that they cover the biweight kernel, which is optimal among non-negative kernels in terms of the asymptotic statistical efficiency for the KDE-based mode estimation.

Point Spread Function Approximation of High-Rank Hessians with Locally Supported Nonnegative Integral Kernels

Point Spread Function Approximation of High-Rank Hessians with Locally Supported Nonnegative Integral Kernels

Effective classification of alzheimer disease based on image tractography framework utilizing GM-ABC-NN

A progressive neurological disorder that leads to memory loss as well as cognitive decline is called Alzheimer’s Disease (AD). Grounded on various factors like symptoms, biomarkers, or disease stages, the classification of AD is done. Yet, the prevailing techniques pose challenges in capturing accurate AD, which results in misdiagnosis or delayed treatment. To overcome these issues, an effective framework for the classification of AD based on image tractography using several algorithms is proposed in this article. Primarily, the dataset comprising AD images undergoes image transformation. Then, to enhance the MRI image quality, the transformed images are pre-processed using Block Matching 3D Filtering (BM3D) and Contrast Limited Adaptive Histogram Equalization (CLAHE) algorithms. Afterward, image tractography followed by brain tissue segmentation is done utilizing the Cauchy-Steiner Weighted Fuzzy C Means (CauSt-FCM) algorithm. Then, by utilizing the Cosine Non-Negative Kernel Regression (CosNNK) algorithm, graph construction is done. Lastly, for the feature selection and validation process, the Premature Singer Archerfish Hunting Optimizer (PreS-AHO) algorithm and Analysis of Variance (ANOVA) analysis are used. Thereafter, the AD stages are classified by using the Gaussian Mixture Alpha-Beta Convolutional Neural Network (GM-ABC-NN) algorithm, The proposed system attains superior accuracy (97.56%), precision (97.55%), and recall (97.57%).

Dispersion analysis of SPH as a way to understand its order of approximation

Smoothed Particle Hydrodynamics (SPH) is a numerical method to solve dynamical partial differential equations (PDE). The basis of the method is a ¡¡kernel-based¿¿ way to compute the spatial derivatives of a function whose values are given in moving irregularly located nodes (Lagrangian particles). Accuracy of the SPH is determined by independent parameters — the shape of the kernel, the kernel size h, the distance between the particles Δx. Constructing high-order SPH-schemes for different types of PDE is a state-of-the-art problem of computational mathematics.For the classical SPH-approximation of one-dimensional hyperbolic equations (isothermal gas dynamics) we found that the order of approximation of smooth solution correlates to the dispersion properties of the method. To this end we analyzed the dispersion relation for the approximation and found analytical representation of the numerical wave phase velocity. Moreover, for the first time, the order of approximation with respect to Δx/h was confirmed in computational experiments on a dynamic problem of sound wave propagation. For two kernels with 2 and 4 continuum derivatives, the second and the fourth order of approximation, respectively, was found. This finding may be generalized as follows. The solution error in the one-dimensional case for a quasi-uniformly located particles has the form O((hλ)η+(Δxh)ξ), where ξ is a parameter determined by the shape of kernel (its smoothness, i.e. the number of continuum derivatives), η is a parameter that does not depend on the shape of kernel (for classical non-negative kernels η=2), λ is the wavelength.Our results indicates that to develop high-order SPH-schemes for hyperbolic equations besides improving the order of approximation with respect to h/λ one need to ensure the order of approximation with respect to Δx/h. To this end kernels of which smoothness is at least 4 are necessary.

Hausdorff–Berezin operators on weighted spaces

The research presented in this paper is a continuation of the recent studies of new classes of integral operators on the unit disc , which are called Hausdorff–Berezin operators. We consider general operators of Hausdorff–Berezin type constructed with an arbitrary positive Radon measure on the unit disc within the framework of weighted spaces with the so‐called Möbius weights. This is a fairly wide class of weights in the unit disc which includes classical radial weights with singularities or zeroes at the boundary. Sufficient conditions for the boundedness of the Hausdorff–Berezin operators in these spaces are obtained. In the case of nonnegative kernels, a boundedness criterion is obtained in . Approximation properties of operators of Hausdorff–Berezin type are also studied. We also discuss some applications.

Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

Abstract In this work, we analyze Merriman, Bence and Osher’s thresholding scheme, a time discretization for mean curvature flow. We restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoğlu and Otto, we show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional convergence results of Otto and one of the authors become unconditional in the two-phase mean convex case. Our results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen.

3PNMF-MKL: A non-negative matrix factorization-based multiple kernel learning method for multi-modal data integration and its application to gene signature detection.

In this current era, biomedical big data handling is a challenging task. Interestingly, the integration of multi-modal data, followed by significant feature mining (gene signature detection), becomes a daunting task. Remembering this, here, we proposed a novel framework, namely, three-factor penalized, non-negative matrix factorization-based multiple kernel learning with soft margin hinge loss (3PNMF-MKL) for multi-modal data integration, followed by gene signature detection. In brief, limma, employing the empirical Bayes statistics, was initially applied to each individual molecular profile, and the statistically significant features were extracted, which was followed by the three-factor penalized non-negative matrix factorization method used for data/matrix fusion using the reduced feature sets. Multiple kernel learning models with soft margin hinge loss had been deployed to estimate average accuracy scores and the area under the curve (AUC). Gene modules had been identified by the consecutive analysis of average linkage clustering and dynamic tree cut. The best module containing the highest correlation was considered the potential gene signature. We utilized an acute myeloid leukemia cancer dataset from The Cancer Genome Atlas (TCGA) repository containing five molecular profiles. Our algorithm generated a 50-gene signature that achieved a high classification AUC score (viz., 0.827). We explored the functions of signature genes using pathway and Gene Ontology (GO) databases. Our method outperformed the state-of-the-art methods in terms of computing AUC. Furthermore, we included some comparative studies with other related methods to enhance the acceptability of our method. Finally, it can be notified that our algorithm can be applied to any multi-modal dataset for data integration, followed by gene module discovery.

Nonconvex low-rank tensor approximation with graph and consistent regularizations for multi-view subspace learning

Multi-view clustering is widely used to improve clustering performance. Recently, the subspace clustering tensor learning method based on Markov chain is a crucial branch of multi-view clustering. Tensor learning is commonly used to apply tensor low-rank approximation to represent the relationships between data samples. However, most of the current tensor learning methods have the following shortcomings: the information of the local graph is not taken into account, the relationships between different views are not shown, and the existing tensor low-rank representation takes a biased tensor rank function for estimation. Therefore, a nonconvex low-rank tensor approximation with graph and consistent regularizations (NLRTGC) model is proposed for multi-view subspace learning. NLRTGC retains the local manifold information through graph regularization, and adopts a consistent regularization between multi-views to keep the diagonal block structure of representation matrices. Furthermore, a nonnegative nonconvex low-rank tensor kernel function is used to replace the existing classical tensor nuclear norm via tensor-singular value decomposition (t-SVD), so as to reduce the deviation from rank. Then, an alternating direction method of multipliers (ADMM) which makes the objective function monotonically non-increasing is proposed to solve NLRTGC. Finally, the effectiveness and superiority of the NLRTGC are shown through abundant comparative experiments with various state-of-the-art algorithms on noisy datasets and real world datasets.

Totally positive kernels, Pólya frequency functions, and their transforms

The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A. M. Whitney’s density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Pólya frequency functions, and Pólya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.

Alexandrov theorem for general nonlocal curvatures: The geometric impact of the kernel

For a general radially symmetric, non-increasing, non-negative kernel h∈Lloc1(Rd), we study the rigidity of measurable sets in Rd with constant nonlocal h-mean curvature. Under a suitable “improved integrability” assumption on h, we prove that these sets are finite unions of equal balls, as soon as they satisfy a natural nondegeneracy condition. Both the radius of the balls and their mutual distance can be controlled from below in terms of suitable parameters depending explicitly on the measure of the level sets of h. In the simplest, common case, in which h is positive, bounded and decreasing, our result implies that any bounded open set or any bounded measurable set with finite perimeter which has constant nonlocal h-mean curvature has to be a ball.

Uniformly complete consistency of frequency polygon estimation for dependent samples and an application

ABSTRACT The frequency polygon estimation, which is based on histogram technique, has a similar convergence rate as those of modern non-negative kernel estimators and the advantage of computational simplicity. This paper will study the uniformly complete consistency and the rates of uniformly complete consistency of frequency polygon estimation for widely orthant dependent samples under some general conditions, which improve and extend the corresponding ones in the literature. As an application, the uniformly complete consistency and the corresponding rates of hazard rate function estimator are also obtained.

Comparison principles for solutions to the fractional differential inequalities with the general fractional derivatives and their applications

In this paper, we discuss some comparison principles for the solutions to the fractional differential inequalities with the general fractional derivatives in the Caputo and Riemann-Liouville senses. These general fractional derivatives are defined as compositions of the first order derivative and a convolution integral with a non-negative and non-increasing kernel. First we prove some estimates for these derivatives acting on the non-negative functions. These estimates are employed for derivation of the comparison principles in several different forms. Finally, we consider an application of the comparison principles for analysis of solutions to the initial-value problems for the fractional differential equations with the general fractional derivatives.

Сompactness of one class of fractional integrating operator with variable upper limit

The paper establishes a characterization of the compactness for fractional operators of a general class, including the Riemann-Liouville, Hadamard and Erdelyi-Kober operators. The paper considers an integral fractional integration operator of Hardy type with nonnegative kernels and a variable limit of integration (a function as the upper limit of integration) and under certain conditions on the kernel, a criterion of the compactness in weighted Lebesgue spaces is obtained for this operator, when the parameters of the spaces satisfy the conditions Moreover, more general results are obtained for the weighted differential inequality of Hardy type on the set of locally absolutely continuous functions that vanish and infinity at the ends of the interval, covering the previously known results, and more precise estimates for the best constant are given. The localization method, Schauder’s theorem, the Kantorovich test, and the theorem on the uniform limit of compact operators were used in the proof of the main theorem. The obtained results of the study the compactness of fractional integration operators can be used in the estimation of solutions of differential equations that model various processes in mathematics. In particular, these results yield new results in the theory of Hardy-type inequalities.

Singular integrals on regular curves in the Heisenberg group

Let H be the first Heisenberg group, and let k∈C∞(H∖{0}) be a kernel which is either odd or horizontally odd, and satisfies|∇Hnk(p)|≤Cn‖p‖−1−n,p∈H∖{0},n≥0. The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel k(p)=∇Hlog⁡‖p‖. We prove that convolution with k, as above, yields an L2-bounded operator on regular curves in H. This extends a theorem of G. David to the Heisenberg group.As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield L2 bounded operators on Lipschitz flags in H. This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li.

Schur-type Banach modules of integral kernels acting on mixed-norm Lebesgue spaces

Schur's test for integral operators states that if a kernel K:X×Y→C satisfies ∫Y|K(x,y)|dν(y)≤C and ∫X|K(x,y)|dμ(x)≤C, then the associated integral operator is bounded from Lp(ν) into Lp(μ), simultaneously for all p∈[1,∞]. We derive a variant of this result which ensures that the integral operator acts boundedly on the (weighted) mixed-norm Lebesgue spaces Lwp,q, simultaneously for all p,q∈[1,∞]. For non-negative integral kernels our criterion is sharp; that is, the integral operator satisfies our criterion if and only if it acts boundedly on all of the mixed-norm Lebesgue spaces.Motivated by this new form of Schur's test, we introduce solid Banach modules Bm(X,Y) of integral kernels with the property that all kernels in Bm(X,Y) map the mixed-norm Lebesgue spaces Lwp,q(ν) boundedly into Lvp,q(μ), for arbitrary p,q∈[1,∞], provided that the weights v,w are m-moderate. Conversely, we show that if A and B are non-trivial solid Banach spaces for which all kernels K∈Bm(X,Y) define bounded maps from A into B, then A and B are related to mixed-norm Lebesgue-spaces, in the sense that (L1∩L∞∩L1,∞∩L∞,1)v↪B and A↪(L1+L∞+L1,∞+L∞,1)1/w for certain weights v,w depending on the weight m used in the definition of Bm.The kernel algebra Bm(X,X) is particularly suited for applications in (generalized) coorbit theory. Usually, a host of technical conditions need to be verified to guarantee that the coorbit space CoΨ(A) associated to a continuous frame Ψ and a solid Banach space A are well-defined and that the discretization machinery of coorbit theory is applicable. As a simplification, we show that it is enough to check that certain integral kernels associated to the frame Ψ belong to Bm(X,X); this ensures that the spaces CoΨ(Lκp,q) are well-defined for all p,q∈[1,∞] and all weights κ compatible with m. Further, if some of these integral kernels have sufficiently small norm, then the discretization theory is also applicable.

Numerical Methods for a Diffusive Class of Nonlocal Operators

In this paper we develop a numerical scheme based on quadratures to approximate solutions of integro-differential equations involving convolution kernels, \(\nu \), of diffusive type. In particular, we assume \(\nu \) is symmetric and exponentially decaying at infinity. We consider problems posed in bounded domains and in \({\mathbb {R}}\). In the case of bounded domains with nonlocal Dirichlet boundary conditions, we show the convergence of the scheme for kernels that have positive tails, but that can take on negative values. When the equations are posed on all of \({\mathbb {R}}\), we show that our scheme converges for nonnegative kernels. Since nonlocal Neumann boundary conditions lead to an equivalent formulation as in the unbounded case, we show that these last results also apply to the Neumann problem.

Large mass minimizers for isoperimetric problems with integrable nonlocal potentials

This paper is concerned with volume-constrained minimization problems derived from Gamow’s liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider a large class of potentials, given by general radial nonnegative kernels which are integrable on Rn, such as Bessel potentials, and study the behavior of the problem for large masses (i.e., volumes). Contrary to the small mass case, where the nonlocal term becomes negligible compared to the perimeter, here the nonlocal term explodes compared to it. However, using the integrability of those kernels, we rewrite the problem as the minimization of the difference between the classical perimeter and a nonlocal perimeter, which converges to a multiple of the classical perimeter as the mass goes to infinity. Renormalizing to a fixed volume, we show that, if the first moment of the kernels is smaller than an explicit threshold, the problem admits minimizers of arbitrarily large mass, which contrasts with the usual case of Riesz potentials. In addition, we prove that, any sequence of minimizers converges to the ball as the mass goes to infinity. Finally, we study the stability of the ball, and show that our threshold on the first moment of the kernels is sharp in the sense that large balls go from stable to unstable. A direct consequence of the instability of large balls above this threshold is that there exist nontrivial compactly supported kernels for which the problems admit minimizers which are not balls, that is, symmetry breaking occurs.

Solvability for a New Class of Moore-Gibson-Thompson Equation with Viscoelastic Memory, Source Terms, and Integral Condition

This paper deals with the existence and uniqueness of solutions of a new class of Moore-Gibson-Thompson equation with respect to the nonlocal mixed boundary value problem, source term, and nonnegative memory kernel. Galerkin’s method was the main used tool for proving our result. This work is a generalization of recent homogenous work.

SOLVABILITY OF THE MOORE–GIBSON–THOMPSON EQUATION WITH VISCOELASTIC MEMORY TERM AND INTEGRAL CONDITION VIA GALERKIN METHOD

We consider the following abstract version of the Moore–Gibson–Thompson (MGT) equation: [Formula: see text] depending on the parameters [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text] is a convex and nonnegative memory kernel. The related energy has been shown to decay exponentially by Boulaaras et al., Math. Meth. Appl. Sci. 42 (2019) 2664–2679; Lasiecka et al., Z. Angew. Math. Phys. 67 (2016) 17. Here, in this original work, the solvability of the abstract version of nonlocal mixed boundary value problem for the MGT equation via Galerkin’s method is discussed.

Fast and precise single-cell data analysis using a hierarchical autoencoder

A primary challenge in single-cell RNA sequencing (scRNA-seq) studies comes from the massive amount of data and the excess noise level. To address this challenge, we introduce an analysis framework, named single-cell Decomposition using Hierarchical Autoencoder (scDHA), that reliably extracts representative information of each cell. The scDHA pipeline consists of two core modules. The first module is a non-negative kernel autoencoder able to remove genes or components that have insignificant contributions to the part-based representation of the data. The second module is a stacked Bayesian autoencoder that projects the data onto a low-dimensional space (compressed). To diminish the tendency to overfit of neural networks, we repeatedly perturb the compressed space to learn a more generalized representation of the data. In an extensive analysis, we demonstrate that scDHA outperforms state-of-the-art techniques in many research sub-fields of scRNA-seq analysis, including cell segregation through unsupervised learning, visualization of transcriptome landscape, cell classification, and pseudo-time inference.