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
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
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
Jeffrey Finkelstein, Hescott, Benjamin · arXiv (Cornell University) · 2016
In the framework of computational complexity and in an effort to define a more natural reduction for problems of equivalence, we investigate the recently introduced kernel reduction, a reduction that operates on each element of a pair independently. This paper details the limitations and uses of kernel reductions. We show that kernel reductions are weaker than many-one reductions and provide conditions under which complete problems exist. Ultimately, the number and size of equivalence classes can dictate the existence of a kernel reduction. We leave unsolved the unconditional existence of a complete problem under polynomial-time kernel reductions for the standard complexity classes.
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2018
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Zhihua Su, Z. Su, P. Qiu · 2009
In the one-dimensional case, we formulate a jump regression model as follows: f(x) = g(x) + p∑ djI(x> sj), for x ∈ [0, 1], i=1 where g is a continuous function, p is the number of jump points, {sj, j = 1, 2,..., p} are the jump positions, and {dj, j = 1, 2,..., p} are the corresponding jump magnitudes. But in reality, our observations will also contain noises. And here for simplicity, we assume the noises ɛi’s are independent and identically distributed random errors with E(ɛi) = 0 and V ar(ɛi) = σ 2 (Xi). But for simplicity, here we just assume that the noise level for all the design points are the same. So we have Yi | Xi = f(Xi) + ɛi, i = 1, 2, · · · , n, where Yi’s are data we can observe. And Xi’s, which are called the design points, are the locations we take observations. 1 One approach to detect jumps is to use a difference polynomial kernel estimation procedure. This procedure estimate the left limit of a design point and also the right limit of a design point by a local polynomial kernel estimator. For an example, if we use a local kernel estimator (polynomial order equals to 0), the estimator will be: MDKE(x) = 1 nhn n∑ i=1
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Hescott, Ben, Finkelstein, Jeffrey · OpenBU/Boston University Institutional Repository (Boston University) · 2015
In this paper, we examine a recently introduced type of effective reduction which applies solely to problems of equivalence or isomorphism: the "kernel reduction". Specifically, we examine reductions among languages in the complexity class consisting of all languages induced by equivalence relations for which membership can be decided by a non-deterministic polynomial time Turing machine. This class is called "NPEq"; the definitions for PEq and coNPEq are analogous. \nWe prove a general theorem which provides a problem which is hard under polynomial time kernel reductions for several classes of equivalence relations, including (Sigma_k)PEq and PSPACEEq. In fact, such a problem is complete for PSPACEEq under polynomial time kernel reductions. We also show that if there is a complete problem under kernel reductions in NPEq, then that problem is also complete under many-one reductions in NP. Finally we use a proof of Ladner's theorem to show that if PEq does not equal NPEq and there are problems in NPEq which are complete under polynomial time kernel reductions then there are NPEq-intermediary problems---problems which are in NPEq, but not complete under kernel reductions and not in PEq.
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Jorge Luis Ojeda Cabrera · 2007
also density.It comprises several different utilities to handle kernel estimators.
Cited by 10Open access
Jeffrey Finkelstein, Benjamin Hescott · 2011
In this paper, we examine a recently introduced type of effective reduction which applies solely to problems of equivalence or isomorphism: the “kernel reduction”. Specifically, we examine reductions among languages in the complexity class consisting of all languages induced by equivalence relations for which membership can be decided by a non-deterministic polynomial time Turing machine. This class is called NPEq; the definitions for PEq and coNPEq are analagous. We prove a general theorem which provides a problem which is hard under polynomial time kernel reductions for several classes of equivalence relations, including ΣkPEq and PSPACEEq. In fact, such a problem is complete for PSPACEEq under polynomial time kernel reductions. We also show that if there is a complete problem under kernel reductions in NPEq, then that problem is also complete under many-one reductions in NP. Finally we use a proof of Ladner’s theorem to show that if PEq = NPEq and there are problems in NPEq which are complete under polynomial time kernel reductions then there are NPEq-intermediary problems—problems which are in NPEq, but not complete under kernel reductions and not in PEq. 1
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Perch Nielsen, Jens, Tanggaard, Carsten · RePEc: Research Papers in Economics
This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it symptotically reduces bias with unchanged variance. A simulation study investigates the finite-sample properties of GPA. The method is tested on local constant and local linear estimators both with and without multiplicative bias correction and with the additive bias correction proposed in Nielsen and Tanggaard (2000). From the simulation experiment we conclude that the global estimator improves the goodness-of-fit. An especially encouraging result is that the bias-correction works well for small samples, where traditional bias reduction methods have a tendency to fail.
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Sadiq H. Abdulhussain, Abd Rahman Ramli, Abir Hussain et al. · 2019
Preprocessing operations of images and video frame sequences are beneficial in computer vision algorithms. For example, smoothing frames is used to eliminate noise; while computing frame gradient in x-direction and y-direction is used for frame feature extraction or for finding frame edges. Such operations involve convolving operators (image kernels) with an image precomputing moments will add extra computation cost to computer vision algorithm. In case of video, the computational time accumulatively increased because of the convolution operation for each frame is performed. To overcome this problem, a mathematical model is established for computing preprocessed frame moments via embedding the operator (image kernel) in the orthogonal polynomial (OP) functions. The experimental results show that the computation time for feature extraction using the proposed method is noticeably reduced in the both trends: image size and moment selection order. The average speed up ratio of the proposed method to traditional method is 3x, 5x, 8x, and 40x for moment selection ratio 100%, 25%, 10%, and 5%, respectively. In addition, the percentage reduction in processing time for small image size is ~ 99% and for large image size is ~ 40%.
Cited by 33
Shawkat Ali, K.A. Smith · Figshare · 2004
Kernel is the heart of kernel based learning. To choose an appropriate parameter for a specific kernel is an important research issue in the data mining area. In this paper, we propose an automatic parameter selection approach for polynomial kernel. The algorithm is tested on support vector machines (SVM). The parameter selection is considered on the basis of prior information of the data distribution and Bayesian inference. The new approach is tested on different sizes of benchmark datasets with binary class problems as well as multi class classification problems.
Cited by 47Open access
Yibo Li, Chao Liu, Senyue Zhang et al. · Journal of Advanced Computational Intelligence and Intelligent Informatics · 2017
Conventional kernel support vector machine (KSVM) has the problem of slow training speed, and single kernel extreme learning machine (KELM) also has some performance limitations, for which this paper proposes a new combined KELM model that build by the polynomial kernel and reproducing kernel on Sobolev Hilbert space. This model combines the advantages of global and local kernel function and has fast training speed. At the same time, an efficient optimization algorithm called cuckoo search algorithm is adopted to avoid blindness and inaccuracy in parameter selection. Experiments were performed on bi-spiral benchmark dataset, Banana dataset, as well as a number of classification and regression datasets from the UCI benchmark repository illustrate the feasibility of the proposed model. It achieves the better robustness and generalization performance when compared to other conventional KELM and KSVM, which demonstrates its effectiveness and usefulness.
Cited by 1Open access
Shao-Bo Lin, Xingping Sun, Zongben Xu et al. · arXiv (Cornell University) · 2015
Polynomial kernel regression is one of the standard and state-of-the-art learning strategies. However, as is well known, the choices of the degree of polynomial kernel and the regularization parameter are still open in the realm of model selection. The first aim of this paper is to develop a strategy to select these parameters. On one hand, based on the worst-case learning rate analysis, we show that the regularization term in polynomial kernel regression is not necessary. In other words, the regularization parameter can decrease arbitrarily fast when the degree of the polynomial kernel is suitable tuned. On the other hand,taking account of the implementation of the algorithm, the regularization term is required. Summarily, the effect of the regularization term in polynomial kernel regression is only to circumvent the " ill-condition" of the kernel matrix. Based on this, the second purpose of this paper is to propose a new model selection strategy, and then design an efficient learning algorithm. Both theoretical and experimental analysis show that the new strategy outperforms the previous one. Theoretically, we prove that the new learning strategy is almost optimal if the regression function is smooth. Experimentally, it is shown that the new strategy can significantly reduce the computational burden without loss of generalization capability.
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Habib Rebei, Nouf Alharbi · Journal of Applied Mathematics and Physics · 2022
In machines learning problems, Support Vector Machine is a method of classification. For non-linearly separable data, kernel functions are a basic ingredient in the SVM technic. In this paper, we briefly recall some useful results on decomposition of RKHS. Based on orthogonal polynomial theory and Mercer theorem, we construct the high power Legendre polynomial kernel on the cube [-1,1]d. Following presentation of the theoretical background of SVM, we evaluate the performance of this kernel on some illustrative examples in comparison with Rbf, linear and polynomial kernels.
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森田 一, Hajime Morita, 高村 大也 et al. · Tokyo Tech Research Repository (Tokyo Institute of Technology) · 2010
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Tyler Chen · The Journal of Chemical Physics · 2023
The kernel polynomial method (KPM) is a powerful numerical method for approximating spectral densities. Typical implementations of the KPM require an a prior estimate for an interval containing the support of the target spectral density, and while such estimates can be obtained by classical techniques, this incurs addition computational costs. We propose a spectrum adaptive KPM based on the Lanczos algorithm without reorthogonalization, which allows the selection of KPM parameters to be deferred to after the expensive computation is finished. Theoretical results from numerical analysis are given to justify the suitability of the Lanczos algorithm for our approach, even in finite precision arithmetic. While conceptually simple, the paradigm of decoupling computation from approximation has a number of practical and pedagogical benefits, which we highlight with numerical examples.
Cited by 3
Abdolreza Shirvani, Hossein Chegini, Saeed Setayeshi et al. · 2009 14th International CSI Computer Conference · 2009
Polynomials are one of the most powerful functions that have been used in many fields of mathematics such as curve fitting and regression. Low order polynomials are desired for their smoothness, good local approximation and interpolation. Being smooth, they can be used to locally approximate almost any derivable function. This means that when linear functions fail in approximation (e.g. where the first order Taylor expansion equals zero) polynomial functions can be used in local approximation, such that one can achieve better estimations at extremums. In this paper, application of polynomial kernel functions in locally linear neurofuzzy models is shown. Using polynomial kernels in local models, better local approximations in prediction of chaotic time series such as Mackey-Glass is achieved, and the capability of the neurofuzzy network is enhanced.
Cited by 3
Arangarajan Vinayagam (12009743), Mohammad Lutfi Othman (9225544), Veerapandiyan Veerasamy (11269145) et al. · Figshare · 2022
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Pål Grønås Drange, Michał Pilipczuk · Algorithmica · 2017
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