Learning an intersection of k halfspaces over a uniform distribution

Authors:
A. Blum;R. Kannan
Affiliations:
Sch. of Comput. Sci., Carnegie Mellon Univ., Pittsburgh, PA, USA;-
Venue:
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
Year:
1993

Citing 0
Cited 2

An Algorithmic Theory of Learning: Robust Concepts and Random Projection

FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
A random-sampling-based algorithm for learning intersections of halfspaces

Journal of the ACM (JACM)

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Abstract

We present a polynomial-time algorithm to learn an intersection of a constant number of halfspaces in n dimensions, over the uniform distribution on an n-dimensional ball. The algorithm we present in fact can learn an intersection of an arbitrary (polynomial) number of halfspaces over this distribution, if the subspace spanned by the normal vectors to the bounding hyperplanes has constant dimension. This generalizes previous results for this distribution, in particular a result of E.B. Baum (1990) who showed how to learn an intersection of 2 halfspaces defined by hyperplanes that pass through the origin (his results in fact held for a variety of symmetric distributions). Our algorithm uses estimates of second moments to find vectors in a low-dimensional "relevant subspace". We believe that the algorithmic techniques studied here may be useful in other geometric learning applications.