Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-4hcbs Total loading time: 0.355 Render date: 2021-11-30T04:03:22.517Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

2 - Kernel-induced vector spaces

from Part I - Machine learning and kernel vector spaces

Published online by Cambridge University Press:  05 July 2014

S. Y. Kung
Affiliation:
Princeton University, New Jersey
Get access

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.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Send book to Kindle

To send this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Kernel-induced vector spaces
  • S. Y. Kung, Princeton University, New Jersey
  • Book: Kernel Methods and Machine Learning
  • Online publication: 05 July 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9781139176224.004
Available formats
×

Send book to Dropbox

To send 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 sending content to Dropbox.

  • Kernel-induced vector spaces
  • S. Y. Kung, Princeton University, New Jersey
  • Book: Kernel Methods and Machine Learning
  • Online publication: 05 July 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9781139176224.004
Available formats
×

Send book to Google Drive

To send 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 sending content to Google Drive.

  • Kernel-induced vector spaces
  • S. Y. Kung, Princeton University, New Jersey
  • Book: Kernel Methods and Machine Learning
  • Online publication: 05 July 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9781139176224.004
Available formats
×