Theoretical Analyses on 2-Norm-Based Multiple Kernel Regressors

Akira TANAKA; Hideyuki IMAI

doi:10.1587/transfun.E100.A.877

Theoretical Analyses on 2-Norm-Based Multiple Kernel Regressors

Akira TANAKA, Hideyuki IMAI

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The solution of the standard 2-norm-based multiple kernel regression problem and the theoretical limit of the considered model space are discussed in this paper. We prove that 1) The solution of the 2-norm-based multiple kernel regressor constructed by a given training data set does not generally attain the theoretical limit of the considered model space in terms of the generalization errors, even if the training data set is noise-free, 2) The solution of the 2-norm-based multiple kernel regressor is identical to the solution of the single kernel regressor under a noise free setting, in which the adopted single kernel is the sum of the same kernels used in the multiple kernel regressor; and it is also true for a noisy setting with the 2-norm-based 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 it is enough to use the single kernel regressors with the sum of given multiple kernels instead of the multiple kernel regressors as long as the 2-norm based criterion is used.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E100-A No.3 pp.877-887

Publication Date: 2017/03/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E100.A.877

Type of Manuscript: PAPER

Category: Neural Networks and Bioengineering

Authors

Akira TANAKA
Hokkaido University
Hideyuki IMAI
Hokkaido University

Keyword

multiple kernel regressor, reproducing kernel Hilbert space, generalization error, 2-norm criterion, 2-norm regularizer

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Akira TANAKA, Hideyuki IMAI, "Theoretical Analyses on 2-Norm-Based Multiple Kernel Regressors" in IEICE TRANSACTIONS on Fundamentals, vol. E100-A, no. 3, pp. 877-887, March 2017, doi: 10.1587/transfun.E100.A.877.
Abstract: The solution of the standard 2-norm-based multiple kernel regression problem and the theoretical limit of the considered model space are discussed in this paper. We prove that 1) The solution of the 2-norm-based multiple kernel regressor constructed by a given training data set does not generally attain the theoretical limit of the considered model space in terms of the generalization errors, even if the training data set is noise-free, 2) The solution of the 2-norm-based multiple kernel regressor is identical to the solution of the single kernel regressor under a noise free setting, in which the adopted single kernel is the sum of the same kernels used in the multiple kernel regressor; and it is also true for a noisy setting with the 2-norm-based 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 it is enough to use the single kernel regressors with the sum of given multiple kernels instead of the multiple kernel regressors as long as the 2-norm based criterion is used.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E100.A.877/_p

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@ARTICLE{e100-a_3_877,
author={Akira TANAKA, Hideyuki IMAI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Theoretical Analyses on 2-Norm-Based Multiple Kernel Regressors},
year={2017},
volume={E100-A},
number={3},
pages={877-887},
abstract={The solution of the standard 2-norm-based multiple kernel regression problem and the theoretical limit of the considered model space are discussed in this paper. We prove that 1) The solution of the 2-norm-based multiple kernel regressor constructed by a given training data set does not generally attain the theoretical limit of the considered model space in terms of the generalization errors, even if the training data set is noise-free, 2) The solution of the 2-norm-based multiple kernel regressor is identical to the solution of the single kernel regressor under a noise free setting, in which the adopted single kernel is the sum of the same kernels used in the multiple kernel regressor; and it is also true for a noisy setting with the 2-norm-based 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 it is enough to use the single kernel regressors with the sum of given multiple kernels instead of the multiple kernel regressors as long as the 2-norm based criterion is used.},
keywords={},
doi={10.1587/transfun.E100.A.877},
ISSN={1745-1337},
month={March},}

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TY - JOUR
TI - Theoretical Analyses on 2-Norm-Based Multiple Kernel Regressors
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 877
EP - 887
AU - Akira TANAKA
AU - Hideyuki IMAI
PY - 2017
DO - 10.1587/transfun.E100.A.877
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E100-A
IS - 3
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - March 2017
AB - The solution of the standard 2-norm-based multiple kernel regression problem and the theoretical limit of the considered model space are discussed in this paper. We prove that 1) The solution of the 2-norm-based multiple kernel regressor constructed by a given training data set does not generally attain the theoretical limit of the considered model space in terms of the generalization errors, even if the training data set is noise-free, 2) The solution of the 2-norm-based multiple kernel regressor is identical to the solution of the single kernel regressor under a noise free setting, in which the adopted single kernel is the sum of the same kernels used in the multiple kernel regressor; and it is also true for a noisy setting with the 2-norm-based 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 it is enough to use the single kernel regressors with the sum of given multiple kernels instead of the multiple kernel regressors as long as the 2-norm based criterion is used.
ER -