Semi-Supervised Learning on Riemannian Manifolds

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Semi-Supervised Learning on Riemannian Manifolds (2004)

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by Mikhail Belkin , Partha Niyogi

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BibTeX

@MISC{Belkin04semi-supervisedlearning,
    author = {Mikhail Belkin and Partha Niyogi},
    title = {Semi-Supervised Learning on Riemannian Manifolds},
    year = {2004}
}

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Abstract

We consider the general problem of utilizing both labeled and unlabeled data to improve classification accuracy. Under the assumption that the data lie on a submanifold in a high dimensional space, we develop an algorithmic framework to classify a partially labeled data set in a principled manner. The central idea of our approach is that classification functions are naturally defined only on the submanifold in question rather than the total ambient space. Using the Laplace-Beltrami operator one produces a basis (the Laplacian Eigenmaps) for a Hilbert space of square integrable functions on the submanifold. To recover such a basis, only unlabeled examples are required. Once such a basis is obtained, training can be performed using the labeled data set. Our algorithm models the manifold using the adjacency graph for the data and approximates the Laplace-Beltrami operator by the graph Laplacian. We provide details of the algorithm, its theoretical justification, and several practical applications for image, speech, and text classification.

Keyphrases

riemannian manifold semi-supervised learning laplace-beltrami operator data set adjacency graph laplacian eigenmaps graph laplacian square integrable function theoretical justification text classification general problem algorithm model data lie high dimensional space unlabeled example total ambient space classification accuracy principled manner hilbert space several practical application central idea algorithmic framework classification function

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