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2 - Kernel-induced vector spaces

from Part I - Machine learning and kernel vector spaces

Published online by Cambridge University Press:  05 July 2014

Summary

Introduction

The notion of kernel-induced vector spaces is the cornerstone of kernel-based machine learning. Generalization of the traditional definition of a similarity metric plays a vital role in facilitating the analysis of complex and big data. It is often necessary to generalize the traditional Euclidean inner product to the more flexible and nonlinear inner products characterized by properly chosen kernel functions. The new inner product leads to a new distance metric, allowing the data analysis to be effectively performed in a higher-dimensional vector space. The topics addressed in this chapter are as follows.

• Section 2.2 introduces Mercer's fundamental theorem stating the necessary and sufficient condition for a function be a Mercer kernel. It will examine several prominent kernel functions, including the polynomial and Gaussian kernel functions.

• Section 2.3 introduces the notion of intrinsic space associated with a kernel function. The intrinsic space is so named because it is independent of the training dataset. The dimension of the space is denoted by J and will be referred to as the intrinsic degree. This degree, whether finite or infinite, dictates the training efficiency and computational cost.

• Section 2.4 introduces a finite-dimensional kernel-induced vector space, known as empirical space, which is jointly determined by the kernel function and the training dataset. The dimension of the empirical space is equal to the data size N. With the LSP condition, both the intrinsic-space and the kernelized learning model will be at our disposal.

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Publisher: Cambridge University Press
Print publication year: 2014

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• Kernel-induced vector spaces
• Book: Kernel Methods and Machine Learning
• Online publication: 05 July 2014
• Chapter DOI: https://doi.org/10.1017/CBO9781139176224.004
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• Kernel-induced vector spaces
• Book: Kernel Methods and Machine Learning
• Online publication: 05 July 2014
• Chapter DOI: https://doi.org/10.1017/CBO9781139176224.004
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

• Kernel-induced vector spaces
• Book: Kernel Methods and Machine Learning
• Online publication: 05 July 2014
• Chapter DOI: https://doi.org/10.1017/CBO9781139176224.004
Available formats
×