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Mercer Large-Scale Kernel Machines from Ridge Function Perspective

Karol Dziedziul, Sergey Kryzhevich, Paweł Wieczyński

TL;DR

This article studies which kernels could be approximated by a sum of products of cosine functions with arguments depending on x and y and presents the obstacles of such an approach.

Abstract

To present Mercer large-scale kernel machines from a ridge function perspective, we recall the results by Lin and Pinkus from {\it Fundamentality of ridge functions}. We consider the main result of the recent paper by Rachimi and Recht, 2008, {\it Random features for large-scale kernel machines} from the Approximation Theory point of view. We study which kernels could be approximated by a sum of products of cosine functions with arguments depending on $x$ and $y$ and present the obstacles of such an approach. The results of this article are applied to Image Processing by procedure "one-vs-rest".

Mercer Large-Scale Kernel Machines from Ridge Function Perspective

TL;DR

This article studies which kernels could be approximated by a sum of products of cosine functions with arguments depending on x and y and presents the obstacles of such an approach.

Abstract

To present Mercer large-scale kernel machines from a ridge function perspective, we recall the results by Lin and Pinkus from {\it Fundamentality of ridge functions}. We consider the main result of the recent paper by Rachimi and Recht, 2008, {\it Random features for large-scale kernel machines} from the Approximation Theory point of view. We study which kernels could be approximated by a sum of products of cosine functions with arguments depending on and and present the obstacles of such an approach. The results of this article are applied to Image Processing by procedure "one-vs-rest".
Paper Structure (7 sections, 9 theorems, 125 equations, 2 figures, 1 table)

This paper contains 7 sections, 9 theorems, 125 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polynomials vanishing on $L(\Omega_S)$
  4. Characterization of polynomials in $\overline{{\mathcal{M}}(\Omega_S)}$
  5. Two application of Universal Approximation Theorem
  6. Criteria of choosing Mercer kernels in learning theory
  7. Experimental results

Key Result

Theorem 2.1

The linear space ${\mathcal{M}}(\Omega)$ is dense in $C(\mathbb{R}^{\tau},\mathbb{R})$ endowed with the topology of uniform convergence on compact sets if and only if the only homogeneous polynomial of $m$ variables which vanishes identically on $L(\Omega)$ is the zero polynomial.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 8 more