Semi-Supervised Learning on Riemannian Manifolds

• Published in: Machine learning Vol. 56; no. 1-3; pp. 209 - 239
• Main Authors: Belkin, Mikhail, Niyogi, Partha
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01.07.2004
• Subjects: Mathematical functions; Studies
• ISSN: 0885-6125; 1573-0565
• DOI: 10.1023/B:MACH.0000033120.25363.1e

Issue Title: Theoretical Advances in Data Clustering 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.[PUBLICATION ABSTRACT]