Learning an Intersection of k Halfspaces over a Uniform Distribution

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 hyper-planes has constant dimension. This generalizes previous results for this distribution, in particular a result of Baum 1] who showed how to learn an intersection of 2 halfspaces deened 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 nd vectors in a low-dimensional \relevant subspace." We believe that the algorithmic techniques studied here may be useful in other geometric learning applications.