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Semi-Supervised Learning$
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Olivier Chapelle, Bernhard Scholkopf, and Alexander Zien

Print publication date: 2006

Print ISBN-13: 9780262033589

Published to MIT Press Scholarship Online: August 2013

DOI: 10.7551/mitpress/9780262033589.001.0001

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Label Propagation and Quadratic Criterion

Label Propagation and Quadratic Criterion

(p.192) (p.193) 11 Label Propagation and Quadratic Criterion
Semi-Supervised Learning

Bengio Yoshua

Delalleau Olivier

Roux Nicolas Le

The MIT Press

This chapter shows how the different graph-based algorithms for semi-supervised learning can be cast into a common framework where one minimizes a quadratic cost criterion whose closed-form solution is found by solving a linear system of size n (total number of data points). The cost criterion naturally leads to an extension of such algorithms to the inductive setting, where one obtains test samples one at a time: the derived induction formula can be evaluated in O(n) time, which is much more efficient than solving again exactly the linear system (which in general costs O(kn2) time for a sparse graph where each data point has k neighbors). This inductive formula is also used to show that when the similarity between points satisfies a locality property, then the algorithms are plagued by the curse of dimensionality, with respect to the dimensionality of an underlying manifold.

Keywords:   graph-based algorithms, semi-supervised learning, quadratic cost criterion, derived induction formula, locality property, dimensionality

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