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Reproducing kernel Banach spaces for machine learning

Haizhang Zhang, Yuesheng Xu, Jun Zhang
2009 2009 International Joint Conference on Neural Networks  
We introduce the notion of reproducing kernel Banach spaces (RKBS) and study special semiinner-product RKBS by making use of semi-inner-products and the duality mapping.  ...  As applications, we develop in the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, support vector machines, and kernel principal component analysis  ...  This paper is motivated from machine learning in Banach spaces. There are advantages of learning in Banach spaces over Hilbert spaces.  ... 
doi:10.1109/ijcnn.2009.5179093 dblp:conf/ijcnn/ZhangXZ09 fatcat:mlvsler6rfakznwatd7d225gci

A Brief Digest on Reproducing Kernel Hilbert Space

Shou-yu TONG, Fu-zhong CONG, Zhi-xia WANG
2017 DEStech Transactions on Computer Science and Engineering  
Reproducing Kernel Hilbert Space (RKHS) is a common used tool in statistics and machine learning to generalize from linear models to non-linear models.  ...  This view is highly related to the kernel methods for regression and classification in the area of machine learning.  ...  Introduction Reproducing Kernel Hilbert Spaces (RKHS) have recently received much attention [1, 2, 3] from the statistics and machine learning researchers, due to the popularity of some machine learning  ... 
doi:10.12783/dtcse/cmee2016/5339 fatcat:cw6e3qwdnbew3ng3pofppol4wu

Generalized Mercer Kernels and Reproducing Kernel Banach Spaces [article]

Yuesheng Xu, Qi Ye
2017 arXiv   pre-print
A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be well-posed and of easy implementation.  ...  This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces (RKHSs).  ...  Machine learning is usually well-posed in reproducing kernel Hilbert spaces (RKHSs). It is desirable to solve learning problems in Banach spaces endowed with certain reproducing kernels.  ... 
arXiv:1412.8663v2 fatcat:s73a2742ujfd7a4a6qbdch2rf4

On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions [article]

Rongrong Lin, Haizhang Zhang, Jun Zhang
2021 arXiv   pre-print
Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation  ...  We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction.  ...  The notion of reproducing kernel Banach spaces was introduced in machine learning in 2009, [46] . Reflexive RKBSs constructed in [46] are the first class of RKBSs with reproducing kernels.  ... 
arXiv:1901.01002v2 fatcat:iypodih4sbhp5h4vhpt5ba3rpu

Transformers are Deep Infinite-Dimensional Non-Mercer Binary Kernel Machines [article]

Matthew A. Wright, Joseph E. Gonzalez
2021 arXiv   pre-print
We prove a new representer theorem for these binary kernel machines with non-Mercer (indefinite, asymmetric) kernels (implying that the functions learned are elements of reproducing kernel Banach spaces  ...  rather than Hilbert spaces), and also prove a new universal approximation theorem showing that the Transformer calculation can learn any binary non-Mercer reproducing kernel Banach space pair.  ...  Section 3 presents the definition of kernel machines on reproducing kernel Banach spaces (RKBS's) that we use to characterize Transformers.  ... 
arXiv:2106.01506v1 fatcat:oin5ri6cxvg7naulfgp3re6fly

A unifying representer theorem for inverse problems and machine learning [article]

Michael Unser
2020 arXiv   pre-print
In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces.  ...  The standard approach for dealing with the ill-posedness of the training problem in machine learning and/or the reconstruction of a signal from a limited number of measurements is regularization.  ...  The author is thankful to Julien Fageot, Shayan Aziznejad, Pakshal Bohra, Quentin Denoyelle and Philippe Thévenaz for helpful comments on the manuscript.  ... 
arXiv:1903.00687v3 fatcat:m636lxldbrcfva4v6a3x2gnaxm

Regularized learning in Banach spaces as an optimization problem: representer theorems

Haizhang Zhang, Jun Zhang
2010 Journal of Global Optimization  
Within the framework of reproducing kernel Banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer  ...  We view regularized learning of a function in a Banach space from its finite samples as an optimization problem.  ...  Following the framework of Tikhonov regularization in machine learning [5, 10, 13, 29, 30, 33, [35] [36] [37] , we let H K be a reproducing kernel Hilbert space (RKHS) on X with the reproducing kernel  ... 
doi:10.1007/s10898-010-9575-z fatcat:3e6j3tnirrbw3biyegcnecbava

Vector-valued Reproducing Kernel Banach Spaces with Applications to Multi-task Learning [article]

Haizhang Zhang, Jun Zhang
2012 arXiv   pre-print
Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBS).  ...  The theory is then applied to multi-task machine learning.  ...  Introduction The purpose of this paper is to establish the notion of vector-valued reproducing kernel Banach spaces and demonstrate its applications to multi-task machine learning.  ... 
arXiv:1111.1037v2 fatcat:kyxgqktqabfu3baoipdr7wie7y

A Unifying Representer Theorem for Inverse Problems and Machine Learning

Michael Unser
2020 Foundations of Computational Mathematics  
In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces.  ...  Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements.  ...  The author is thankful to Julien Fageot, Shayan Aziznejad, Pakshal Bohra, Quentin Denoyelle, and Philippe Thévenaz for helpful comments on the manuscript.  ... 
doi:10.1007/s10208-020-09472-x fatcat:lxgpt4bop5gazfz7z4cwny232u

Multi-task Learning in Vector-valued Reproducing Kernel Banach Spaces with the ℓ^1 Norm [article]

Rongrong Lin, Guohui Song, Haizhang Zhang
2019 arXiv   pre-print
In order to provide a kernel method for this model, we construct a class of vector-valued reproducing kernel Banach spaces with the ℓ^1 norm.  ...  Targeting at sparse multi-task learning, we consider regularization models with an ℓ^1 penalty on the coefficients of kernel functions.  ...  Introduction Reproducing kernel Banach spaces (RKBSs) and their applications have attracted a lot of attention in machine learning community [8, 12, 13, 14, 23, 24, 29, 30, 32, 33] .  ... 
arXiv:1901.01036v1 fatcat:73wwznrgvvguhnxauy5uuwcjae

Solving support vector machines in reproducing kernel Banach spaces with positive definite functions

Gregory E. Fasshauer, Fred J. Hickernell, Qi Ye
2015 Applied and Computational Harmonic Analysis  
The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.  ...  In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert  ...  Introduction The theory and practice of kernel-based methods is a fast growing research area. They have been used for both scattered data approximation and machine learning.  ... 
doi:10.1016/j.acha.2014.03.007 fatcat:h5b4zw5zsza77l42vtqazmlxti

Probability Inequalities for Kernel Embeddings in Sampling without Replacement

Markus Schneider
2016 International Conference on Artificial Intelligence and Statistics  
Furthermore, we derive probability inequalities for Banach space valued martingales in the setting of sampling without replacement.  ...  The kernel embedding of distributions is a popular machine learning technique to manipulate probability distributions and is an integral part of numerous applications.  ...  In the following, we introduce concepts and notation required for the understanding of reproducing kernel Hilbert spaces and the kernel embedding.  ... 
dblp:conf/aistats/Schneider16 fatcat:xww77gfhubbp3kogl6yiankixq

Vector-valued Reproducing Kernel Banach Spaces with Group Lasso Norms [article]

Liangzhi Chen, Haizhang Zhang, Jun Zhang
2019 arXiv   pre-print
Aiming at a mathematical foundation for kernel methods in coefficient regularization for multi-task learning, we investigate theory of vector-valued reproducing kernel Banach spaces (RKBS) with L_p,1-norms  ...  The corresponding kernels that are admissible for the construction are discussed.  ...  Learning in reproducing kernel Hilbert spaces (RKHSs) have received considerable attentions over the past few decades in machine learning [3] , [30] , [31] , statistical learning [4] , [40] and stochastic  ... 
arXiv:1903.00819v1 fatcat:pgzwaydn4jexln62m7xypoiurq

Sparse regularized learning in the reproducing kernel banach spaces with the \begin{document}$ \ell^1 $\end{document} norm

Ying Lin, ,School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China, Rongrong Lin, Qi Ye, ,School of Data and Computer Science, Sun Yat-sen University, Guangzhou, Guangdong 510006, China
2019 Mathematical Foundations of Computing  
Finally, we perform numerical experiments for synthetic data and real-world benchmark data in the reproducing kernel Banach spaces with the 1 norm and the reproducing kernel Hilbert spaces both with Laplacian  ...  We present a sparse representer theorem for regularization networks in a reproducing kernel Banach space with the 1 norm by the theory of convex analysis.  ...  The reproducing kernel Hilbert spaces (RKHSs) have been viewed as ideal spaces for kernel-based learning algorithms [7, 17, 22] .  ... 
doi:10.3934/mfc.2020020 fatcat:zfxgb2pq2zfh3n5cfz6pysu3rm

Reproducing Kernel Banach Spaces with the l1 Norm II: Error Analysis for Regularized Least Square Regression [article]

Guohui Song, Haizhang Zhang
2011 arXiv   pre-print
Following this direction, we illustrate how reproducing kernel Banach spaces with the l1 norm can be applied to improve the learning rate estimate of l1-regularization in machine learning.  ...  Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically.  ...  Introduction A class of reproducing kernel Banach spaces (RKBS) with the ℓ 1 norm that satisfies the linear representer theorem was recently constructed in [14] .  ... 
arXiv:1101.4439v2 fatcat:6ug7pylbvnforggvtxkako4foi
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