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A Reproducing Kernel Hilbert Space Framework for Information-Theoretic Learning

Jian-Wu Xu, A.R.C. Paiva, Il Park, J.C. Principe
2008 IEEE Transactions on Signal Processing  
This paper provides a functional analysis perspective of information-theoretic learning (ITL) by defining bottom-up a reproducing kernel Hilbert space (RKHS) uniquely determined by the symmetric nonnegative  ...  Index Terms-Cross-information potential, information-theoretic learning (ITL), kernel function, probability density function, reproducing kernel Hilbert space (RKHS).  ...  Then is said to be a reproducing kernel Hilbert space with reproducing kernel .  ... 
doi:10.1109/tsp.2008.2005085 fatcat:tn3bj75wdrdaxkupvbc4ug3nyi

Kernel Mean Embedding of Distributions: A Review and Beyond

Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, Bernhard Schölkopf
2017 Foundations and Trends® in Machine Learning  
A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference.  ...  The basic idea behind this framework is to map distributions into a reproducing kernel Hilbert space (RKHS) in which the whole arsenal of kernel methods can be extended to probability measures.  ...  The basic idea behind this framework is to map distributions into a reproducing kernel Hilbert space (RKHS) in which the whole arsenal of kernel methods can be extended to probability measures.  ... 
doi:10.1561/2200000060 fatcat:vgmsbodozngltpzy6c2idxnx34

Ensemble and Multiple Kernel Regressors: Which Is Better?

Akira TANAKA, Hirofumi TAKEBAYASHI, Ichigaku TAKIGAWA, Hideyuki IMAI, Mineichi KUDO
2015 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
, ensemble kernel regressor, multiple kernel regressor, generalization error, reproducing kernel Hilbert spaces  ...  In this paper, we introduce a unified framework for evaluating kernel regressors with multiple kernels.  ...  The Hilbert space H that has a reproducing kernel is called a reproducing kernel Hilbert space (RKHS).  ... 
doi:10.1587/transfun.e98.a.2315 fatcat:yu4zitjj55d75nr56tpg7ac2pa

Learning vocal tract variables with multi-task kernels

Hachem Kadri, Emmanuel Duflos, Philippe Preux
2011 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)  
This paper presents a multi-task kernel based method aimed at learning Vocal Tract (VT) variables from the Mel-Frequency Cepstral Coefficients (MFCCs).  ...  Index Terms-Multi-task learning, matrix-valued kernel, vocal tract variables, acoustic-to-articulatory inversion.  ...  [14] who showed how Hilbert spaces of vector-valued functions [15] and matrix-valued reproducing kernels [16] can be used as a theoretical framework to develop nonlinear multi-task learning methods  ... 
doi:10.1109/icassp.2011.5946917 dblp:conf/icassp/KadriDP11 fatcat:2eq7kdvgovehzbxgt652xvir3a

Extended Analyses for an Optimal Kernel in a Class of Kernels with an Invariant Metric [chapter]

Akira Tanaka, Ichigaku Takigawa, Hideyuki Imai, Mineichi Kudo
2012 Lecture Notes in Computer Science  
In our previous work, we discussed a class of kernels forming a nested class of reproducing kernel Hilbert spaces with an invariant metric and proved that the kernel corresponding to the smallest reproducing  ...  Learning based on kernel machines is widely known as a powerful tool for various fields of information science such as pattern recognition and regression estimation.  ...  This work was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 24500001.  ... 
doi:10.1007/978-3-642-34166-3_38 fatcat:67vmjigwsbdgxcx2ecwwrwk5ti

Functional learning through kernels [article]

Stephane Canu
2009 arXiv   pre-print
To illustrate this framework some examples of such reproducing sets and kernels are given.  ...  Based on these principles an original theory is developed generalizing the notion of reproduction kernel Hilbert space to non hilbertian sets.  ...  This work also benefits from comments and discussion with NATO ASI on Learning Theory and Practice students in Leuven.  ... 
arXiv:0910.1013v1 fatcat:w47ldigvwrbgzpe72ekpcopsxu

The Matrix Hilbert Space and Its Application to Matrix Learning [article]

Yunfei Ye
2017 arXiv   pre-print
In addition, we extend the reproducing kernel Hilbert space (RKHS) to reproducing kernel matrix Hilbert space (RKMHS) and propose an equivalent condition of the space uses of the certain kernel function  ...  Here, we present a new framework, namely matrix Hilbert space to perform a matrix inner product space when data observations are represented as matrices.  ...  We are grateful to Dong Han and Lynn Kuo for our discussions.  ... 
arXiv:1706.08110v2 fatcat:cryrojsplrbyhc4meawhh2r3ty

Integrated kernels and their properties

Akira Tanaka, Hideyuki Imai, Mineichi Kudo, Masaaki Miyakoshi
2007 Pattern Recognition  
By using a kernel, an unknown target is represented by a function that belongs to a reproducing kernel Hilbert space (RKHS) corresponding to the kernel.  ...  Some numerical experiments show that the unresolved problem of finding a good parameter can be neglected. keyword kernel, reproducing kernel Hilbert space, projection learning, parameter integration  ...  Takigawa for their valuable comments. The authors would also like to thank Dr. Kitamura for his useful comments regarding the style of the revised manuscript.  ... 
doi:10.1016/j.patcog.2007.02.014 fatcat:3vmhwvxzgzg33hbwcgcjeezcle

On the PAC-Bayes Bound Calculation based on Reproducing Kernel Hilbert Space

Li Tang, Hua Yu, Xiu-Jun GONG
2013 Applied Mathematics & Information Sciences  
In this paper, we formulated the concept space as Reproducing Kernel Hilbert Space (RKHS) using the kernel method.  ...  PAC-Bayes risk bound combining Bayesian theory and structure risk minimization for stochastic classifiers has been considered as a framework for deriving some of the tightest generalization bounds.  ...  RH is so-called the Reproducing Kernel Hilbert Space(RKHS). In this sense, K is called the reproducing kernel.  ... 
doi:10.12785/amis/070253 fatcat:vilrumatljb57bhmk4hebsvz2y

Preface to the special issue on analysis in machine learning and data science

Andreas Chirstmann, ,Department of Mathematics Stochastics University of Bayreuth 95440 Bayreuth, Germany, Qiang Wu, Ding-Xuan Zhou, ,Department of Mathematical Sciences Middle Tennessee State University Murfreesboro, TN 37132, USA, ,School of Data Science and Department of Mathematics City University of Hong Kong 83 Tat Chee Avenue, Kowloon Hong Kong, China
2020 Communications on Pure and Applied Analysis  
Rastogi considered least squares based Tikhonov regularization schemes in vector valued reproducing kernel Hilbert spaces for certain nonlinear ill-posed operator equation with a nonlinear forward operator  ...  Hu studied the gradient descent method for the maximum correntropy criterion associated with reproducing kernel Hilbert spaces and proved the near optimal convergence rates.  ... 
doi:10.3934/cpaa.2020171 fatcat:tu3hfj2rvrgl7ih65dzae5i2xe

Optimal Kernel in a Class of Kernels with an Invariant Metric [chapter]

Akira Tanaka, Hideyuki Imai, Mineichi Kudo, Masaaki Miyakoshi
2008 Lecture Notes in Computer Science  
Learning based on kernel machines is widely known as a powerful tool for various fields of information science such as pattern recognition and regression estimation.  ...  In this paper, we consider a class of kernels that forms a monotonic classes of reproducing kernel Hilbert spaces with an invariant metric and show that the kernel corresponding to the smallest reproducing  ...  This work was partially supported by Grant-in-Aid No. 18700001 for Young Scientists (B) from the Ministry of Education, Culture, Sports and Technology of Japan.  ... 
doi:10.1007/978-3-540-89689-0_57 fatcat:7nn2w5rhlzcezakpb4mbbwbqcu

Information Theoretic Learning with Infinitely Divisible Kernels [article]

Luis G. Sanchez Giraldo, Jose C. Principe
2013 arXiv   pre-print
In this paper, we develop a framework for information theoretic learning based on infinitely divisible matrices.  ...  The proposed formulation avoids the plug in estimation of density and brings along the representation power of reproducing kernel Hilbert spaces.  ...  Part of the research effort in this context has pointed out connections to reproducing kernel Hilbert spaces [2] .  ... 
arXiv:1301.3551v6 fatcat:6wdzcoyma5dt5f356vpugkxgii

A Relationship Between Generalization Error and Training Samples in Kernel Regressors

Akira Tanaka, Hideyuki Imai, Mineichi Kudo, Masaaki Miyakoshi
2010 2010 20th International Conference on Pattern Recognition  
In our previous work, we gave a framework to evaluate the first component.  ...  One is a distance between an unknown true function and an adopted model space.  ...  ACKNOWLEDGMENT This work was partially supported by Grant-in-Aid No.21700001 for Young Scientists (B) from the Ministry of Education, Culture, Sports and Technology of Japan.  ... 
doi:10.1109/icpr.2010.351 dblp:conf/icpr/TanakaIKM10 fatcat:u7eg2dcubrge5ddc7muwulhfie

Stability of Multi-Task Kernel Regression Algorithms [article]

Julien Audiffren
2013 arXiv   pre-print
We study the stability properties of nonlinear multi-task regression in reproducing Hilbert spaces with operator-valued kernels.  ...  Such kernels, a.k.a. multi-task kernels, are appropriate for learning prob- lems with nonscalar outputs like multi-task learning and structured out- put prediction.  ...  Let (Ω, F , P) be a probability space, X a Polish space, Y a (possibly infinite-dimensional) separable Hilbert space, H a separable Reproducing Kernel Hilbert Space (RKHS) ⊂ Y X with K its reproducing  ... 
arXiv:1306.3905v1 fatcat:ewlknblwajagvjxqkdtdcbvro4

Theoretical Analyses on 2-Norm-Based Multiple Kernel Regressors

Akira TANAKA, Hideyuki IMAI
2017 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
regressor, reproducing kernel Hilbert space, generalization error, 2-norm criterion, 2-norm regularizer  ...  The first result motivates us to develop a novel framework for the multiple kernel regression problems which yields a better solution close to the theoretical limit, and the second result implies that  ...  Acknowledgment This work was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 24500001 and 16K05264.  ... 
doi:10.1587/transfun.e100.a.877 fatcat:ltrrugxotjfhjkrrrhlay6sk6q
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