Machine Learning

, Volume 56, Issue 1, pp 209–239

Semi-Supervised Learning on Riemannian Manifolds

  • Mikhail Belkin
  • Partha Niyogi

DOI: 10.1023/B:MACH.0000033120.25363.1e

Cite this article as:
Belkin, M. & Niyogi, P. Machine Learning (2004) 56: 209. doi:10.1023/B:MACH.0000033120.25363.1e


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.

semi-supervised learningmanifold learninggraph regularizationlaplace operatorgraph laplacian
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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Mikhail Belkin
    • 1
  • Partha Niyogi
    • 1
  1. 1.Department of Computer ScienceUniversity of ChicagoChicagoUSA