Homogeneous Kernel Research Articles

Weighted oscillation and variation inequalities for singular integrals and commutators satisfying Hörmander type conditions

This paper is devoted to investigating the weighted L p -mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish the weighted type (p, p) estimates for 1 < p < ∞ and the weighted weak type (1, 1) estimate for the oscillation and variation operators of singular integrals with kernels satisfying certain Hormander type conditions, which contain the Riesz transforms, singular integrals with more general homogeneous kernels satisfying the Lipschitz conditions and the classical Dini’s conditions as model examples. Meanwhile, we also obtain the weighted L p -boundeness for such operators associated to the family of commutators generated by the singular integrals above with BMO(Rd)-functions.

On Hardy-type integral inequalities with the gamma function

By means of real analysis and weight functions, we obtain a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters. The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions and some cases of homogeneous kernel.

A time-driven constant-number Monte Carlo method for the GPU-simulation of particle breakage based on weighted simulation particles

Monte Carlo (MC) simulations based on weighted particles offer novel and more precise techniques for the solution of the population balance equation for particulate systems. A recent constant-number approach named stochastic weighted algorithm (SWA) (Lee et al. (2015), J. Comput. Phys. (303) 1–18) has been developed, which renders the breakage of a simulation particle by an alteration of its properties, without the creation of novel simulation particles. The theoretic justification of the general formulation for all possible SWAs is limited to binary breakage kernels. We present a novel approach for the derivation of the properties of the MC particles representing fragments, which is applicable for all sorts of breakage kernels. This general scheme encompasses the already introduced SWA schemes, especially a number-based (SWA1, named NBS in this paper) and volume-based (SWA2, named VBS in this paper) breakage scheme, and it makes novel formulations possible: the low volume scheme (LVS), which renders preferably lower fragment sizes, and the combination of LVS with the NBS (LVS-NBS) or VBS (LVS-VBS). The implementation of these breakage schemes in the context of a GPU-based time-driven method is presented and the gained results are validated by comparison with results of the analytic solutions of a homogeneous binary breakage kernel. It is found, that the SWA methods (NBS and VBS) are only able to render large particle sizes, and that LVS, NBS-LVS and VBS-LVS are able to render the whole spectrum of particle sizes. Smaller noise levels are found for VBS and specific VBS-LVS schemes, making both more suitable for prolonged simulations than the other presented methods.

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L 2(w), we obtain a bound that is quadratic in A 2 constant $${\left[ w \right]_{{A_2}}}$$ . We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.

Binary Embedding with Additive Homogeneous Kernels

Binary embedding transforms vectors in Euclidean space into the vertices of Hamming space such that Hamming distance between binary codes reflects a particular distance metric. In machine learning, the similarity metrics induced by Mercer kernels are frequently used, leading to the development of binary embedding with Mercer kernels (BE-MK) where the approximate nearest neighbor search is performed in a reproducing kernel Hilbert space (RKHS). Kernelized locality-sensitive hashing (KLSH), which is one of the representative BE-MK, uses kernel PCA to embed data points into a Euclidean space, followed by the random hyperplane binary embedding. In general, it works well when the query and data points in the database follow the same probability distribution. The streaming data environment, however, continuously requires KLSH to update the leading eigenvectors of the Gram matrix, which can be costly or hard to carry out in practice. In this paper we present a completely randomized binary embedding to work with a family of additive homogeneous kernels, referred to as BE-AHK. The proposed algorithm is easy to implement, built on Vedaldi and Zisserman's work on explicit feature maps for additive homogeneous kernels. We show that our BE-AHK is able to preserve kernel values by developing an upper- and lower-bound on its Hamming distance, which guarantees to solve approximate nearest neighbor search efficiently. Numerical experiments demonstrate that BE-AHK actually yields similarity-preserving binary codes in terms of additive homogeneous kernels and is superior to existing methods in case that training data and queries are generated from different distributions. Moreover, in cases where a large code size is allowed, the performance of BE-AHK is comparable to that of KLSH in general cases.

Isotropic wave turbulence with simplified kernels: Existence, uniqueness, and mean-field limit for a class of instantaneous coagulation-fragmentation processes

The isotropic 4-wave kinetic equation is considered in its weak formulation using model (simplified) homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting where the kernels have a rate of growth at most linear. We also consider finite stochastic particle systems undergoing instantaneous coagulation-fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit).

Learning deep kernels in the space of dot product polynomials

Recent literature has shown the merits of having deep representations in the context of neural networks. An emerging challenge in kernel learning is the definition of similar deep representations. In this paper, we propose a general methodology to define a hierarchy of base kernels with increasing expressiveness and combine them via multiple kernel learning (MKL) with the aim to generate overall deeper kernels. As a leading example, this methodology is applied to learning the kernel in the space of Dot-Product Polynomials (DPPs), that is a positive combination of homogeneous polynomial kernels (HPKs). We show theoretical properties about the expressiveness of HPKs that make their combination empirically very effective. This can also be seen as learning the coefficients of the Maclaurin expansion of any definite positive dot product kernel thus making our proposed method generally applicable. We empirically show the merits of our approach comparing the effectiveness of the kernel generated by our method against baseline kernels (including homogeneous and non homogeneous polynomials, RBF, etc...) and against another hierarchical approach on several benchmark datasets.

Evaluation of a rapid hybrid spectral-spatial domain 3D forward-modeling approach for loop-loop electromagnetic induction quadrature data acquired in low-induction-number environments

When exploring subsurface environments using electromagnetic (EM) induction (EMI) tools, approximate forward-modeling methods based on a homogeneous half-space kernel have been extensively evaluated in the past. For large-scale exploration methods, such as magnetotellurics, marine EM, airborne EM, transient EM, and large offset loop-loop harmonic EM, such forward-modeling approaches are limited because the kernel depends strongly on the subsurface distribution of electrical conductivity. However, the response of small portable EMI loop-loop sensors applied in a low-induction number (LIN) context are known to be more linearly related to the true distribution of electrical conductivity. Thus, data collected using such sensors are more adapted to an approximate forward-modeling with a conductivity-independent kernel. We have evaluated the bias of such an approximate modeling for the case of portable multiconfiguration system measurements in 1D, 2D, and 3D contexts. Our result shows that the approximate approach tends to underestimate the conductivity of more conductive targets but is able to reproduce the right structural information. Compared with previous algorithms presented in the literature, we solved the approximate forward-modeling problem in the hybrid spectral-spatial domain to speed up the computation. Considering the level of accuracy in structural modeling as well as the computational efficiency of our hybrid spectral-spatial approach, we conclude that this method is especially suitable for near-surface, large-scale mapping applications in LIN environments as typically encountered in soil sciences and archaeological studies. For such applications, our approach can be implemented in rapid multichannel deconvolution procedures.

Application of a homogenous patch mean kernel with within-class collaborative representation for hyperspectral imagery classification

ABSTRACTSpectral–spatial-based classification methods demonstrate satisfying performance for hyperspectral imagery (HSI) classification. In this letter, in order to make full use of spectral and contexture information with simultaneously considering within-class information, we propose a new algorithm for HSI classification based on within-class collaborative representation and column generation (CG) strategy. The proposed accelerated homogeneous patch mean kernel (HPMK) can automatically assign a homogeneous patch for the target sample and represent the similarities between training set and assigned homogeneous patch in kernel feature space based on CG strategy. Further, for including intra-class information and improve classification efficiency, within-class collaborative representation classification (WCRC) is incorporated into new feature space to enhance the classification performance. Experiments on two real HSI data sets demonstrate that the proposed algorithm presents satisfying results in terms of classification accuracy and efficiency.

Some sharp inequalities for integral operators with homogeneous kernel

For the third author the research was supported by grant No. 15-01-02732 of the Russian Fund of Basic Research

On a Hardy-Hilbert-type inequality with a general homogeneous kernel

By the method of weight coefficients and techniques of real analysis, a Hardy-Hilbert-type inequality with a general homogeneous kernel and a best possible constant factor is given. The equivalent forms, the operator expressions with the norm, the reverses and some particular examples are also considered.

C*-algebra of integral operators with homogeneous kernels and oscillating coefficients

We consider the C*-algebra generated by multidimensional integral operators with (−n)th-order homogeneous kernels and by the operators of multiplication by oscillating coefficients of the form |x|iα. For this algebra, we construct an operator symbolic calculus and obtain necessary and sufficient conditions for the Fredholm property of an operator in terms of this calculus.

Hashing via Efficient Addictive Kernel for Logistics Image Classification

In this paper, fast image search with efficient additive kernels and kernel locality-sensitive hashing has been proposed. As to hold the kernel functions, recent work has probed methods to create locality-sensitive hashing , which guarantee our approach’s linear time, however existing methods still do not solve the problem of locality-sensitive hashing (LSH) and indirectly sacrifice the loss in accuracy of search results in order to allow fast queries. To improve the search accuracy, we show how to apply explicit feature maps into the homogeneous kernels, which help in feature transformation and combine it with kernel locality-sensitive hashing. We prove our method on several large datasets, and illustrate that it improve the accuracy relative to commonly used methods and make the task of object classification, content-based retrieval more fast and accurate.

Mixed-Kernels Constructions of Polar Codes

Mixed kernels of polar codes are mapping functions having inputs of different alphabet sizes that are used to construct polar coding scheme. These schemes are constructed by incorporating several (homogeneous) kernels, each one over different alphabet size. In this paper, the idea of mixed-kernels construction is introduced and analyzed. An asymptotic analysis of the proposed scheme shows that its polarization properties are strongly related to the ones of the constituent kernels. Simulation of finite length instances of the scheme indicate their advantages both in error correction performance and complexity compared to the known polar coding structures.

Some sharp inequalities for multidimensional integral operators with homogeneous kernel: an overview and new results

One goal of this paper is to point out the fact that a big number of inequalities provedfrom time to time in journal publications, both one-dimensional and multi-dimensional, are particularcases of ...

Salient pairwise spatio-temporal interest points for real-time activity recognition

Real-time Human action classification in complex scenes has applications in various domains such as visual surveillance, video retrieval and human robot interaction. While, the task is challenging due to computation efficiency, cluttered backgrounds and intro-variability among same type of actions. Spatio-temporal interest point (STIP) based methods have shown promising results to tackle human action classification in complex scenes efficiently. However, the state-of-the-art works typically utilize bag-of-visual words (BoVW) model which only focuses on the word distribution of STIPs and ignore the distinctive character of word structure. In this paper, the distribution of STIPs is organized into a salient directed graph, which reflects salient motions and can be divided into a time salient directed graph and a space salient directed graph, aiming at adding spatio-temporal discriminant to BoVW. Generally speaking, both salient directed graphs are constructed by labeled STIPs in pairs. In detail, the “directional co-occurrence” property of different labeled pairwise STIPs in same frame is utilized to represent the time saliency, and the space saliency is reflected by the “geometric relationships” between same labeled pairwise STIPs across different frames. Then, new statistical features namely the Time Salient Pairwise feature (TSP) and the Space Salient Pairwise feature (SSP) are designed to describe two salient directed graphs, respectively. Experiments are carried out with a homogeneous kernel SVM classifier, on four challenging datasets KTH, ADL and UT-Interaction. Final results confirm the complementary of TSP and SSP, and our multi-cue representation TSP + SSP + BoVW can properly describe human actions with large intro-variability in real-time.

Large-scale brain dynamics: effect of connectivity resolution

Large-scale brain dynamics recently started to be modeled numerically based on both heterogeneous large-scale networks build from diffusion MRI, that is, a connectome, and local homogeneous connectivity kernel representing intracortical synaptic connections. However, topological properties of a connectome can significantly change with resolution and parcellation [1]. Furthermore, the sampling of the cerebral surfaces, resulting in a geometric model, and, in this way, the local connectivity kernel play crucial roles in the formation of spatial patterns on the cerebral surfaces as well as on the sensor level (e.g., EEG electrodes) by a forward calculation [2]. However, the effect of sampling and parcellation on modeling brain dynamics has not been studied so far. Here, we investigate qualitatively and quantitatively: (i) how different parcellation resolutions affect the dynamics of the network; and (ii) how the local connectivity affects the network dynamics. To do so, we used the neuroinformatics platform for large-scale brain simulations, called The Virtual Brain (TVB) [3] and developed a preprocessing pipeline to incorporate experimental data (e.g., structural MRI, diffusion-weighted MRI) in TVB [4]. We prepared ten individual models based on ten randomly selected subjects from the Human Connectome Project dataset [5]. For each individual model we performed simulations under two conditions during rest: (i) noise driven, using a bistable neural mass model, and (ii) after stimulation, using an excitable neural mass model. We investigated the effect of heterogeneous and homogeneous connectivity on large-scale brain dynamics by different numbers of regions in the parcellation (70 to 2240) and by varying the local connectivity coupling strength. To introduce experimental data (i.e., structural and diffusion MRI) into TVB we tackled issues such as surface downsampling (for achieving moderate simulation times) and mapping between surface and parcellation (to consistently use heterogeneous and homogeneous connectivity) by developing the Surface and Connectivity Reconstruction with an Imaging Pipeline for TVB Simulations, short SCRIPTS [4]. When considering slow dynamics, the major fiber bundles best reflected in the coarsest parcellation appeared to be mainly responsible for the emergence of the network attractors with limited changes over different parcellations and different local coupling strengths. For fast dynamics, new qualitative solutions appeared, but only in the presence of delays. Figure 1 A. Spatial attractors. For each value of the global (G) or local (G_local) coupling parameter, correlation with in-strength (blue points) and with s-core (red points) for ten different initial conditions. The blue square indicates the critical interval. ...

Sharp weighted bounds for multilinear fractional maximal type operators with rough kernels

In this paper, we will give the weighted bounds for multilinear fractional maximal type operators $\mathcal{M}_{\Omega,\alpha}$ with rough homogeneous kernels. We obtain a mixed $A_{(\vec{P},q)}-A_\infty$ bound and a $A_{\vec{P}}$ type estimate for $\mathcal{M}_{\Omega,\alpha}$. As an application, we give an almost sharp estimate for the multilinear fractional integral operator with rough kernels $\mathcal{I}_{\Omega,\alpha}$.

Hopf bifurcation in a diffusive Lotka–Volterra type system with nonlocal delay effect

The dynamics of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and Dirichlet boundary conditions is investigated in this paper. The existence and multiplicity of spatially nonhomogeneous steady-state solutions are obtained by means of Lyapunov–Schmidt reduction. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain.

Extension of the Nakajima-Zwanzig approach to multitime correlation functions of open systems

We extend the Nakajima-Zwanzig projection operator technique to the determination of multitime correlation functions of open quantum systems. The correlation functions are expressed in terms of certain multitime homogeneous and inhomogeneous memory kernels for which suitable equations of motion are derived. We show that under the condition of finite memory times these equations can be used to determine the memory kernels by employing an exact stochastic unraveling of the full system-environment dynamics. The approach thus allows to combine exact stochastic methods, feasible for short times, with long-time master equation simulations. The applicability of the method is demonstrated by numerical simulations of 2D-spectra for a donor-acceptor model, and by comparison of the results with those obtained from the reduced hierarchy equations of motion. We further show that the formalism is also applicable to the time evolution of a periodically driven two-level system initially in equilibrium with its environment.