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Academic literature on the topic 'Approximation de Nyström'

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Published: 25 May 2024

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Journal articles on the topic "Approximation de Nyström"

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Ding, Lizhong, Yong Liu, Shizhong Liao, Yu Li, Peng Yang, Yijie Pan, Chao Huang, Ling Shao, and Xin Gao. "Approximate Kernel Selection with Strong Approximate Consistency." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 3462–69. http://dx.doi.org/10.1609/aaai.v33i01.33013462.

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Kernel selection is fundamental to the generalization performance of kernel-based learning algorithms. Approximate kernel selection is an efficient kernel selection approach that exploits the convergence property of the kernel selection criteria and the computational virtue of kernel matrix approximation. The convergence property is measured by the notion of approximate consistency. For the existing Nyström approximations, whose sampling distributions are independent of the specific learning task at hand, it is difficult to establish the strong approximate consistency. They mainly focus on the quality of the low-rank matrix approximation, rather than the performance of the kernel selection criterion used in conjunction with the approximate matrix. In this paper, we propose a novel Nyström approximate kernel selection algorithm by customizing a criterion-driven adaptive sampling distribution for the Nyström approximation, which adaptively reduces the error between the approximate and accurate criteria. We theoretically derive the strong approximate consistency of the proposed Nyström approximate kernel selection algorithm. Finally, we empirically evaluate the approximate consistency of our algorithm as compared to state-of-the-art methods.
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Wang, Ling, Hongqiao Wang, and Guangyuan Fu. "Multi-Nyström Method Based on Multiple Kernel Learning for Large Scale Imbalanced Classification." Computational Intelligence and Neuroscience 2021 (June 13, 2021): 1–11. http://dx.doi.org/10.1155/2021/9911871.

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Extensions of kernel methods for the class imbalance problems have been extensively studied. Although they work well in coping with nonlinear problems, the high computation and memory costs severely limit their application to real-world imbalanced tasks. The Nyström method is an effective technique to scale kernel methods. However, the standard Nyström method needs to sample a sufficiently large number of landmark points to ensure an accurate approximation, which seriously affects its efficiency. In this study, we propose a multi-Nyström method based on mixtures of Nyström approximations to avoid the explosion of subkernel matrix, whereas the optimization to mixture weights is embedded into the model training process by multiple kernel learning (MKL) algorithms to yield more accurate low-rank approximation. Moreover, we select subsets of landmark points according to the imbalance distribution to reduce the model’s sensitivity to skewness. We also provide a kernel stability analysis of our method and show that the model solution error is bounded by weighted approximate errors, which can help us improve the learning process. Extensive experiments on several large scale datasets show that our method can achieve a higher classification accuracy and a dramatical speedup of MKL algorithms.
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Zhang, Kai, and James T. Kwok. "Density-Weighted Nyström Method for Computing Large Kernel Eigensystems." Neural Computation 21, no. 1 (January 2009): 121–46. http://dx.doi.org/10.1162/neco.2009.11-07-651.

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The Nyström method is a well-known sampling-based technique for approximating the eigensystem of large kernel matrices. However, the chosen samples in the Nyström method are all assumed to be of equal importance, which deviates from the integral equation that defines the kernel eigenfunctions. Motivated by this observation, we extend the Nyström method to a more general, density-weighted version. We show that by introducing the probability density function as a natural weighting scheme, the approximation of the eigensystem can be greatly improved. An efficient algorithm is proposed to enforce such weighting in practice, which has the same complexity as the original Nyström method and hence is notably cheaper than several other alternatives. Experiments on kernel principal component analysis, spectral clustering, and image segmentation demonstrate the encouraging performance of our algorithm.
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Díaz de Alba, Patricia, Luisa Fermo, and Giuseppe Rodriguez. "Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules." Numerische Mathematik 146, no. 4 (November 18, 2020): 699–728. http://dx.doi.org/10.1007/s00211-020-01163-7.

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AbstractThis paper is concerned with the numerical approximation of Fredholm integral equations of the second kind. A Nyström method based on the anti-Gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces. The Nyström interpolants corresponding to the Gauss and the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation, under suitable assumptions which are easily verified for a particular weight function. Hence, an error estimate is available, and the accuracy of the solution can be improved by approximating it by an averaged Nyström interpolant. The effectiveness of the proposed approach is illustrated through different numerical tests.
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Rudi, Alessandro, Leonard Wossnig, Carlo Ciliberto, Andrea Rocchetto, Massimiliano Pontil, and Simone Severini. "Approximating Hamiltonian dynamics with the Nyström method." Quantum 4 (February 20, 2020): 234. http://dx.doi.org/10.22331/q-2020-02-20-234.

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Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nyström method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.
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Trokicić, Aleksandar, and Branimir Todorović. "Constrained spectral clustering via multi–layer graph embeddings on a grassmann manifold." International Journal of Applied Mathematics and Computer Science 29, no. 1 (March 1, 2019): 125–37. http://dx.doi.org/10.2478/amcs-2019-0010.

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Abstract We present two algorithms in which constrained spectral clustering is implemented as unconstrained spectral clustering on a multi-layer graph where constraints are represented as graph layers. By using the Nystrom approximation in one of the algorithms, we obtain time and memory complexities which are linear in the number of data points regardless of the number of constraints. Our algorithms achieve superior or comparative accuracy on real world data sets, compared with the existing state-of-the-art solutions. However, the complexity of these algorithms is squared with the number of vertices, while our technique, based on the Nyström approximation method, has linear time complexity. The proposed algorithms efficiently use both soft and hard constraints since the time complexity of the algorithms does not depend on the size of the set of constraints.
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Cai, Difeng, and Panayot S. Vassilevski. "Eigenvalue Problems for Exponential-Type Kernels." Computational Methods in Applied Mathematics 20, no. 1 (January 1, 2020): 61–78. http://dx.doi.org/10.1515/cmam-2018-0186.

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AbstractWe study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate {\lvert\lambda_{k}-\lambda_{k,h}\rvert\leq C_{k}h^{2}} in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, {\lambda_{k}} and {\lambda_{k,h}} are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that {|\lambda^{(G)}_{k,h}-\lambda^{(N)}_{k,h}|\leq Ch^{2}}, where {\lambda^{(G)}_{k,h}} and {\lambda^{(N)}_{k,h}} denote the kth largest eigenvalues computed by Galerkin and Nyström methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.
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He, Li, and Hong Zhang. "Kernel K-Means Sampling for Nyström Approximation." IEEE Transactions on Image Processing 27, no. 5 (May 2018): 2108–20. http://dx.doi.org/10.1109/tip.2018.2796860.

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Wang, Shiyuan, Lujuan Dang, Guobing Qian, and Yunxiang Jiang. "Kernel recursive maximum correntropy with Nyström approximation." Neurocomputing 329 (February 2019): 424–32. http://dx.doi.org/10.1016/j.neucom.2018.10.064.

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Laguardia, Anna Lucia, and Maria Grazia Russo. "A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains." Mathematics 11, no. 23 (December 3, 2023): 4859. http://dx.doi.org/10.3390/math11234859.

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This paper is devoted to the numerical treatment of two-dimensional Fredholm integral equations, defined on general curvilinear domains of the plane. A Nyström method, based on a suitable Gauss-like cubature formula, recently proposed in the literature is proposed. The convergence, stability and good conditioning of the method are proved in suitable subspaces of continuous functions of Sobolev type. The cubature formula, on which the Nyström method is constructed, has an error that behaves like the best polynomial approximation of the integrand function. Consequently, it is also shown how the Nyström method inherits this property and, hence, the proposed numerical strategy is fast when the involved known functions are smooth. Some numerical examples illustrate the efficiency of the method, also in comparison with other methods known in the literature.
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Dissertations / Theses on the topic "Approximation de Nyström"

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Cherfaoui, Farah. "Echantillonnage pour l'accélération des méthodes à noyaux et sélection gloutonne pour les représentations parcimonieuses." Electronic Thesis or Diss., Aix-Marseille, 2022. http://www.theses.fr/2022AIXM0256.

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Les contributions de cette thèse se divisent en deux parties. Une première partie dédiée à l’accélération des méthodes à noyaux et une seconde à l'optimisation sous contrainte de parcimonie. Les méthodes à noyaux sont largement connues et utilisées en apprentissage automatique. Toutefois, la complexité de leur mise en œuvre est élevée et elles deviennent inutilisables lorsque le nombre de données est grand. Nous proposons dans un premier temps une approximation des Ridge Leverage Scores. Nous utilisons ensuite ces scores pour définir une distribution de probabilité pour le processus d'échantillonnage de la méthode de Nyström afin d’accélérer les méthodes à noyaux. Nous proposons dans un second temps un nouveau framework basé sur les noyaux, permettant de représenter et de comparer les distributions de probabilités discrètes. Nous exploitons ensuite le lien entre notre framework et la Maximum Mean Discrepancy pour proposer une approximation précise et peu coûteuse de cette dernière. La deuxième partie de cette thèse est consacrée à l’optimisation avec contrainte de parcimonie pour l’optimisation de signaux et l’élagage de forêts aléatoires. Tout d’abord, nous prouvons sous certaines conditions sur la cohérence du dictionnaire, les propriétés de reconstruction et de convergence de l’algorithme Frank-Wolfe. Ensuite, nous utilisons l'algorithme OMP pour réduire la taille de forêts aléatoires et ainsi réduire la taille nécessaire pour son stockage. La forêt élaguée est constituée d’un sous-ensemble d’arbres de la forêt initiale sélectionnés et pondérés par OMP de manière à minimiser son erreur empirique de prédiction
The contributions of this thesis are divided into two parts. The first part is dedicated to the acceleration of kernel methods and the second to optimization under sparsity constraints. Kernel methods are widely known and used in machine learning. However, the complexity of their implementation is high and they become unusable when the number of data is large. We first propose an approximation of Ridge leverage scores. We then use these scores to define a probability distribution for the sampling process of the Nyström method in order to speed up the kernel methods. We then propose a new kernel-based framework for representing and comparing discrete probability distributions. We then exploit the link between our framework and the maximum mean discrepancy to propose an accurate and fast approximation of the latter. The second part of this thesis is devoted to optimization with sparsity constraint for signal optimization and random forest pruning. First, we prove under certain conditions on the coherence of the dictionary, the reconstruction and convergence properties of the Frank-Wolfe algorithm. Then, we use the OMP algorithm to reduce the size of random forests and thus reduce the size needed for its storage. The pruned forest consists of a subset of trees from the initial forest selected and weighted by OMP in order to minimize its empirical prediction error
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Li, Jun 1977. "A computational model for the diffusion coefficients of DNA with applications." Thesis, 2010. http://hdl.handle.net/2152/ETD-UT-2010-05-1098.

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The sequence-dependent curvature and flexibility of DNA is critical for many biochemically important processes. However, few experimental methods are available for directly probing these properties at the base-pair level. One promising way to predict these properties as a function of sequence is to model DNA with a set of base-pair parameters that describe the local stacking of the different possible base-pair step combinations. In this dissertation research, we develop and study a computational model for predicting the diffusion coefficients of short, relatively rigid DNA fragments from the sequence and the base-pair parameters. We focus on diffusion coefficients because various experimental methods have been developed to measure them. Moreover, these coefficients can also be computed numerically from the Stokes equations based on the three-dimensional shape of the macromolecule. By comparing the predicted diffusion coefficients with experimental measurements, we can potentially obtain refined estimates of various base-pair parameters for DNA. Our proposed model consists of three sub-models. First, we consider the geometric model of DNA, which is sequence-dependent and controlled by a set of base-pair parameters. We introduce a set of new base-pair parameters, which are convenient for computation and lead to a precise geometric interpretation. Initial estimates for these parameters are adapted from crystallographic data. With these parameters, we can translate a DNA sequence into a curved tube of uniform radius with hemispherical end caps, which approximates the effective hydrated surface of the molecule. Second, we consider the solvent model, which captures the hydrodynamic properties of DNA based on its geometric shape. We show that the Stokes equations are the leading-order, time-averaged equations in the particle body frame assuming that the Reynolds number is small. We propose an efficient boundary element method with a priori error estimates for the solution of the exterior Stokes equations. Lastly, we consider the diffusion model, which relates our computed results from the solvent model to relevant measurements from various experimental methods. We study the diffusive dynamics of rigid particles of arbitrary shape which often involves arbitrary cross- and self-coupling between translational and rotational degrees of freedom. We use scaling and perturbation analysis to characterize the dynamics at time scales relevant to different classic experimental methods and identify the corresponding diffusion coefficients. In the end, we give rigorous proofs for the convergence of our numerical scheme and show numerical evidence to support the validity of our proposed models by making comparisons with experimental data.
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Book chapters on the topic "Approximation de Nyström"

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Hutchings, Matthew, and Bertrand Gauthier. "Local Optimisation of Nyström Samples Through Stochastic Gradient Descent." In Machine Learning, Optimization, and Data Science, 123–40. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-25599-1_10.

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AbstractWe study a relaxed version of the column-sampling problem for the Nyström approximation of kernel matrices, where approximations are defined from multisets of landmark points in the ambient space; such multisets are referred to as Nyström samples. We consider an unweighted variation of the radial squared-kernel discrepancy (SKD) criterion as a surrogate for the classical criteria used to assess the Nyström approximation accuracy; in this setting, we discuss how Nyström samples can be efficiently optimised through stochastic gradient descent. We perform numerical experiments which demonstrate that the local minimisation of the radial SKD yields Nyström samples with improved Nyström approximation accuracy in terms of trace, Frobenius and spectral norms.
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Fu, Zhouyu. "Optimal Landmark Selection for Nyström Approximation." In Neural Information Processing, 311–18. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-12640-1_38.

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Li, Hongyu, and Lin Zhang. "Dynamic Subspace Update with Incremental Nyström Approximation." In Computer Vision – ACCV 2010 Workshops, 384–93. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22819-3_39.

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Zhang, Huaxiang, Zhichao Wang, and Linlin Cao. "Fast Nyström for Low Rank Matrix Approximation." In Advanced Data Mining and Applications, 456–64. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-35527-1_38.

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Frammartino, Carmelina. "A Nyström Method for Solving a Boundary Value Problem on [0, ∞)." In Approximation and Computation, 311–25. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-6594-3_20.

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Jia, Hongjie, Liangjun Wang, and Heping Song. "Large-Scale Spectral Clustering with Stochastic Nyström Approximation." In IFIP Advances in Information and Communication Technology, 26–34. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-46931-3_3.

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Allouch, Chafik, Ikram Hamzaoui, and Driss Sbibih. "Richardson Extrapolation of Nyström Method Associated with a Sextic Spline Quasi-Interpolant." In Mathematical and Computational Methods for Modelling, Approximation and Simulation, 105–19. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-94339-4_5.

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Yun, Jeong-Min, and Seungjin Choi. "Nyström Approximations for Scalable Face Recognition: A Comparative Study." In Neural Information Processing, 325–34. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-24958-7_38.

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Conference papers on the topic "Approximation de Nyström"

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Giffon, Luc, Stephane Ayache, Thierry Artieres, and Hachem Kadri. "Deep Networks with Adaptive Nyström Approximation." In 2019 International Joint Conference on Neural Networks (IJCNN). IEEE, 2019. http://dx.doi.org/10.1109/ijcnn.2019.8851711.

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Zhang, Kai, Ivor W. Tsang, and James T. Kwok. "Improved Nyström low-rank approximation and error analysis." In the 25th international conference. New York, New York, USA: ACM Press, 2008. http://dx.doi.org/10.1145/1390156.1390311.

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Mathur, Anant, Sarat Moka, and Zdravko Botev. "Column Subset Selection and Nyström Approximation via Continuous Optimization." In 2023 Winter Simulation Conference (WSC). IEEE, 2023. http://dx.doi.org/10.1109/wsc60868.2023.10407416.

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Münch, Maximilian, Katrin Sophie Bohnsack, Alexander Engelsberger, Frank-Michael Schleif, and Thomas Villmann. "Sparse Nyström Approximation for Non-Vectorial Data Using Class-informed Landmark Selection." In ESANN 2023 - European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning. Louvain-la-Neuve (Belgium): Ciaco - i6doc.com, 2023. http://dx.doi.org/10.14428/esann/2023.es2023-136.

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Patel, Raajen, Tom Goldstein, Eva Dyer, Azalia Mirhoseini, and Richard Baraniuk. "Deterministic Column Sampling for Low-Rank Matrix Approximation: Nyström vs. Incomplete Cholesky Decomposition." In Proceedings of the 2016 SIAM International Conference on Data Mining. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2016. http://dx.doi.org/10.1137/1.9781611974348.67.

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Lee, Jieun, and Yoonsik Choe. "Graph-Regularized Fast Low-Rank Matrix Approximation Using The NystrÖM Method for Clustering." In 2018 IEEE 28th International Workshop on Machine Learning for Signal Processing (MLSP). IEEE, 2018. http://dx.doi.org/10.1109/mlsp.2018.8517034.

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Dereziński, Michał, Rajiv Khanna, and Michael W. Mahoney. "Improved Guarantees and a Multiple-descent Curve for Column Subset Selection and the Nystrom Method (Extended Abstract)." In Thirtieth International Joint Conference on Artificial Intelligence {IJCAI-21}. California: International Joint Conferences on Artificial Intelligence Organization, 2021. http://dx.doi.org/10.24963/ijcai.2021/647.

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The Column Subset Selection Problem (CSSP) and the Nystrom method are among the leading tools for constructing interpretable low-rank approximations of large datasets by selecting a small but representative set of features or instances. A fundamental question in this area is: what is the cost of this interpretability, i.e., how well can a data subset of size k compete with the best rank k approximation? We develop techniques which exploit spectral properties of the data matrix to obtain improved approximation guarantees which go beyond the standard worst-case analysis. Our approach leads to significantly better bounds for datasets with known rates of singular value decay, e.g., polynomial or exponential decay. Our analysis also reveals an intriguing phenomenon: the cost of interpretability as a function of k may exhibit multiple peaks and valleys, which we call a multiple-descent curve. A lower bound we establish shows that this behavior is not an artifact of our analysis, but rather it is an inherent property of the CSSP and Nystrom tasks. Finally, using the example of a radial basis function (RBF) kernel, we show that both our improved bounds and the multiple-descent curve can be observed on real datasets simply by varying the RBF parameter.
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Ahmed, Hesham Ibrahim, Wan Qun, Ding Xue-ke, and Zhou Zhi-ping. "Squared distance matrix completion through Nystrom approximation." In 2016 22nd Asia-Pacific Conference on Communications (APCC). IEEE, 2016. http://dx.doi.org/10.1109/apcc.2016.7581449.

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Hou, Bo-Jian, Lijun Zhang, and Zhi-Hua Zhou. "Storage Fit Learning with Unlabeled Data." In Twenty-Sixth International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/256.

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By using abundant unlabeled data, semi-supervised learning approaches have been found very useful in various tasks. Existing approaches, however, neglect the fact that the storage available for the learning process is different under different situations, and thus, the learning approaches should be flexible subject to the storage budget limit. In this paper, we focus on graph-based semi-supervised learning and propose two storage fit learning approaches which can adjust their behaviors to different storage budgets. Specifically, we utilize techniques of low-rank matrix approximation to find a low-rank approximator of the similarity matrix so as to reduce the space complexity. The first approach is based on stochastic optimization, which is an iterative approach that converges to the optimal low-rank approximator globally. The second approach is based on Nystrom method, which can find a good low-rank approximator efficiently and is suitable for real-time applications. Experiments on classification tasks show that the proposed methods can fit dynamically different storage budgets and obtain good performances in different scenarios.
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Patel, Lokendra Singh, Suman Sana, and S. P. Ghrera. "Efficient Nystrom method for low rank approximation and error analysis." In 2015 Third International Conference on Image Information Processing (ICIIP). IEEE, 2015. http://dx.doi.org/10.1109/iciip.2015.7414831.

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