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Machine Learning

, Volume 65, Issue 1, pp 79–94 | Cite as

Kernels as features: On kernels, margins, and low-dimensional mappings

  • Maria-Florina Balcan
  • Avrim Blum
  • Santosh Vempala
Article

Abstract

Kernel functions are typically viewed as providing an implicit mapping of points into a high-dimensional space, with the ability to gain much of the power of that space without incurring a high cost if the result is linearly-separable by a large margin γ. However, the Johnson-Lindenstrauss lemma suggests that in the presence of a large margin, a kernel function can also be viewed as a mapping to a low-dimensional space, one of dimension only \(\tilde{O}(1/\gamma^2)\). In this paper, we explore the question of whether one can efficiently produce such low-dimensional mappings, using only black-box access to a kernel function. That is, given just a program that computes K(x,y) on inputs x,y of our choosing, can we efficiently construct an explicit (small) set of features that effectively capture the power of the implicit high-dimensional space? We answer this question in the affirmative if our method is also allowed black-box access to the underlying data distribution (i.e., unlabeled examples). We also give a lower bound, showing that if we do not have access to the distribution, then this is not possible for an arbitrary black-box kernel function; we leave as an open problem, however, whether this can be done for standard kernel functions such as the polynomial kernel. Our positive result can be viewed as saying that designing a good kernel function is much like designing a good feature space. Given a kernel, by running it in a black-box manner on random unlabeled examples, we can efficiently generate an explicit set of \(\tilde{O}(1/\gamma^2)\) features, such that if the data was linearly separable with margin γ under the kernel, then it is approximately separable in this new feature space.

Keywords

Kernel Function Mach Learn Target Function Large Margin Polynomial Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Maria-Florina Balcan
    • 1
  • Avrim Blum
    • 1
  • Santosh Vempala
    • 2
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburgh
  2. 2.Department of MathematicsMITCambridge

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