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About 10,000 papers for “Kernel method”

kernel method

2024

Citation: 'kernel method' in the IUPAC Compendium of Chemical Terminology, 5th ed.; International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.11460 • License: The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International for individual terms. Requests for commercial usage of the compendium should be directed to IUPAC.

Cited by 0

CHAPTER 5. THE KERNEL METHOD 137 Chapter 5 The Kernel Method

2012

Before we can mine data, it is important to first find a suitable data representation that facilitates data analysis. For example, for complex data like text, sequences, images, and so on, we must typically extract or construct a set ofattributesor features, so that we may represent the data instances as multivariate vectors. That is, given a data instance x (e.g., a sequence), we need to find a mapping φ, so that φ(x) is the vector representation of x. Evenwhentheinputdataisanumeric data matrix, if we wish to discover non-linear relationships amongtheattributes, then a non-linear mapping φ may be used, so that φ(x) represents a vector in the corresponding high-dimensional space comprising non-linear attributes. We use the term input space to refer to the data space for the input data x, andfeature space to refer to the space of mapped vectors φ(x). Thus, given a set of data objects or instances xi, andgivenamappingfunctionφ, wecantransformthemintofeature vectors φ(xi), whichthenallowsustoanalyzecomplexdatainstancesvianumeric analysis methods. Example 5.1 (Sequence-based Features): Consider a dataset of DNA sequences over the alphabet Σ = {A, C, G, T}. One simple feature space is to represent each sequence in terms of the probability distribution over symbols in Σ. That is, given a sequence x with length |x | = m, themappingintofeature space is given as φ(x) ={P (A),P(C),P(G),P(T)} where P (s) = ns

Cited by 0

The Kernel Polynomial Method

Weisse, Alexander, Wellein, Gerhard, Alvermann, Andreas et al. · arXiv (Cornell University) · 2005

Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed matter physics. In this article we review basic properties and recent developments of Chebyshev expansion based algorithms and the Kernel Polynomial Method. Characterized by a resource consumption that scales linearly with the problem dimension these methods enjoyed growing popularity over the last decade and found broad application not only in physics. Representative examples from the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems we discuss in detail. In addition, we illustrate how the Kernel Polynomial Method is successfully embedded into other numerical techniques, such as Cluster Perturbation Theory or Monte Carlo simulation.

Cited by 4Open access

Kernel Probability Estimation Method

Jensen, Greg · Figshare · 2015

Visualization of the kernel probability estimation procedure. <strong>Left.</strong> Correct responses (in green) and incorrect responses (in purple) are treated as independent time series, in which each is assigned a Gaussian distribution with an optimized bandwidth. This replaces the observed response times (vertical lines) with Gaussian density functions. <strong>Middle.</strong> The Gaussian kernels are summed to yield kernel estimates of the instantaneous rate of each event over time. <strong>Right.</strong> The relative rate of events is calculated for each time point. In addition, the credible interval for the estimate is shown as a dashed line.

Cited by 0Open access

Hybrid kernel polynomial method

Irfan, Muhammad, Sathish R. Kuppuswamy, Varjas, Daniel et al. · Zenodo (CERN European Organization for Nuclear Research) · 2019

The kernel polynomial method allows to sample overall spectral properties of a quantum system, while sparse diagonalization provides accurate information about a few important states. We present a method combining these two approaches without loss of performance or accuracy. We apply this hybrid kernel polynomial method to improve the computation of thermodynamic quantities and the construction of perturbative effective models, in a regime where neither of the methods is sufficient on its own. To achieve this we develop a perturbative kernel polynomial method to compute arbitrary order series expansions of expectation values. We demonstrate the efficiency of our approach on three examples: the calculation of supercurrent and inductance in a Josephson junction, the interaction of spin qubits defined in a two dimensional electron gas, and the calculation of the effective band structure in a realistic model of a semiconductor nanowire.

Cited by 0Open access

The Heat Kernel Method

J. J. Duistermaat · Birkhäuser Boston eBooks · 2011

The goal of the heat kernel method is to express (2.40) as an integral over the fixed point set M γ in M of the transformation γ. Here M γ = M if γ is the identity. The method is based on the following observations about arbitrary elliptic differential operators D, acting on sections of a smooth vector bundle F over a compact manifold M, which admits a a direct sum decomposition F = F + ⊕ F − . In our case, F = E ⊗ L, with the splitting F ± = E ± ⊗ L, and D is the spin-c Dirac operator. For the required facts about trace class operators, see for instance Hörmander [42, Sec. 19.1], or Duistermaat [19].

Cited by 0

Kernel Probability Estimation Method

Greg Jensen · Figshare · 2015

Visualization of the kernel probability estimation procedure. <strong>Left.</strong> Correct responses (in green) and incorrect responses (in purple) are treated as independent time series, in which each is assigned a Gaussian distribution with an optimized bandwidth. This replaces the observed response times (vertical lines) with Gaussian density functions. <strong>Middle.</strong> The Gaussian kernels are summed to yield kernel estimates of the instantaneous rate of each event over time. <strong>Right.</strong> The relative rate of events is calculated for each time point. In addition, the credible interval for the estimate is shown as a dashed line.

Cited by 0Open access

Hybrid kernel polynomial method

Muhammad Irfan, Sathish R. Kuppuswamy, Dániel Varjas et al. · arXiv (Cornell University) · 2019

The kernel polynomial method allows to sample overall spectral properties of a quantum system, while sparse diagonalization provides accurate information about a few important states. We present a method combining these two approaches without loss of performance or accuracy. We apply this hybrid kernel polynomial method to improve the computation of thermodynamic quantities and the construction of perturbative effective models, in a regime where neither of the methods is sufficient on its own. To achieve this we develop a perturbative kernel polynomial method to compute arbitrary order series expansions of expectation values. We demonstrate the efficiency of our approach on three examples: the calculation of supercurrent and inductance in a Josephson junction, the interaction of spin qubits defined in a two dimensional electron gas, and the calculation of the effective band structure in a realistic model of a semiconductor nanowire.

Cited by 2Open access

Kernel SIMPLE: Positron Lifetime Image Reconstruction Using Kernel Method

Bo Huang, Jinyi Qi · 2024

Ortho-positronium (o-Ps) lifetime has the potential to become a biomarker for quantifying oxygen concentration and micro-structural properties of human tissue in vivo. Several model-based lifetime image reconstruction methods have been developed. The most computationally efficient method is the Statistical Image Reconstruction of Positron Lifetime via Time-Weighting (SIMPLE) method. It can achieve a spatial resolution comparable to that of the standard positron emission tomography (PET) images with a computational cost equivalent to two standard PET image reconstructions. However, the method lacks effective regularization to suppress noise in the lifetime images. To address this issue, here we developed a kernel-based SIMPLE method. The kernel matrix is first constructed using either dynamic activity images or lifetime-thresholded images and then utilized to represent both the activity image and the lifetime-weighted image in the SIMPLE method. Computer simulations show the kernel SIMPLE method can substantially reduce noise in lifetime images compared to the original SIMPLE method. The dynamic-image-based kernel matrix demonstrates better noise-suppression capability compared to the lifetime-thresholded kernel matrix.

Cited by 0

Fuzzy clustering using kernel method

Daoqiang Zhang, Songcan Chen · 2005

Classical fuzzy C -means (FCM) clustering is performed in the input space, given the desired number of clusters. Although it has proven effective for spherical data, it fails when the data structure of input patterns is non-spherical and complex. In this paper, we present a novel kernel-based fuzzy C-means clustering algorithm (KFCM). Its basic idea is to transform implicitly the input data into a higher dimensional feature space via a nonlinear map, which increases greatly possibility of linear separability of the patterns in the feature space, then perform FCM in the feature space. Another good attribute of KFCM is that it can automatically estimate the number of clusters in the dataset. The experimental results show that the proposed method has better performance in the Ring dataset.

Cited by 111

Kernel Current Source Density Method

Jan Potworowski, Wit Jakuczun, Szymon Łęski et al. · Neural Computation · 2011

Local field potentials (LFP), the low-frequency part of extracellular electrical recordings, are a measure of the neural activity reflecting dendritic processing of synaptic inputs to neuronal populations. To localize synaptic dynamics, it is convenient, whenever possible, to estimate the density of transmembrane current sources (CSD) generating the LFP. In this work, we propose a new framework, the kernel current source density method (kCSD), for nonparametric estimation of CSD from LFP recorded from arbitrarily distributed electrodes using kernel methods. We test specific implementations of this framework on model data measured with one-, two-, and three-dimensional multielectrode setups. We compare these methods with the traditional approach through numerical approximation of the Laplacian and with the recently developed inverse current source density methods (iCSD). We show that iCSD is a special case of kCSD. The proposed method opens up new experimental possibilities for CSD analysis from existing or new recordings on arbitrarily distributed electrodes (not necessarily on a grid), which can be obtained in extracellular recordings of single unit activity with multiple electrodes.

Cited by 94

Multivariate wavelet kernel regression method

Samir Touzani, Daniel Busby · HAL (Le Centre pour la Communication Scientifique Directe) · 2011

The purpose of this paper is to introduce a new penalized multivariate nonparametric regression method, in the framework of wavelet decomposition. We call this method the wavelet kernel ANOVA (WK-ANOVA), which is a wavelet based reproducing kernel Hilbert space (RKHS) method with the penalty equal to the sum of blockwise RKHS norms. This method does not require design points to be equispaced or of dyadic size thus making high-dimensional wavelet estimation feasible. We also introduce a new iterative shrinkage algorithm to solve the nonegative garrote optimization problem resulting in the variable selection step. Numerical experiments on several test functions show that the WK-ANOVA provides competitive results compared to other standard methods.

Cited by 2Open access

Regularized reproducing kernel particle method

Huy Anh Nguyen, Satoyuki Tanaka · Computational Particle Mechanics · 2025

We study the quasi-consistent reproducing kernel particle method (QC-RKPM), which was recently introduced by Huang and Wei (Comput Mech 70(6):1211–1239, 2022). QC-RKPM (Huang and Wei 2022) was initially proposed to address the kernel instability arising in plate analysis when a quadratic basis is employed. Although QC-RKPM shows potential in overcoming this challenge, its formulation has not been fully derived, and the conditions ensuring the invertibility of the moment matrices remain unexplored. In this work, we present a detailed derivation of QC-RKPM within the framework of the moving least-squares RKPM and identify the requirements for moment matrix invertibility. In addition, we demonstrate that QC-RKPM can effectively mitigate the singularity caused by the linear dependence between RK basis functions and the shell surface function. The conditions under which moment matrices are non-singular in this case are established. A series of numerical examples is presented to verify the theoretical properties and effectiveness of QC-RKPM.

Cited by 0Open access

A Riemann-Stein Kernel Method

Alessandro Barp, Chris J. Oates, Emilio Porcu et al. · arXiv (Cornell University) · 2018

This paper proposes and studies a numerical method for approximation of posterior expectations based on interpolation with a Stein reproducing kernel. Finite-sample-size bounds on the approximation error are established for posterior distributions supported on a compact Riemannian manifold, and we relate these to a kernel Stein discrepancy (KSD). Moreover, we prove in our setting that the KSD is equivalent to Sobolev discrepancy and, in doing so, we completely characterise the convergence-determining properties of KSD. Our contribution is rooted in a novel combination of Stein's method, the theory of reproducing kernels, and existence and regularity results for partial differential equations on a Riemannian manifold.

Cited by 2Open access

A kernel method for unsupervised . . .

Christoph Lippert, Oliver Stegle, Zoubin Ghahramani et al. · 2009

Network inference is the problem of inferring edges between a set of real-world objects, for instance, interactions between pairs of proteins in bioinformatics. Current kernelbased approaches to this problem share a set of common features: (i) they are supervised and hence require labeled training data; (ii) edges in the network are treated as mutually independent and hence topological properties are largely ignored; (iii) they lack a statistical interpretation. We argue that these common assumptions are often undesirable for network inference, and propose (i) an unsupervised kernel method (ii) that takes the global structure of the network into account and (iii) is statistically motivated. We show that our approach can explain commonly used heuristics in statistical terms. In experiments on social networks, different variants of our method demonstrate appealing predictive performance.

Cited by 0