  We study the approximate nearest neighbour method for cost-sensitive
classification on low-dimensional manifolds embedded within a high-dimensional
feature space. We determine the minimax learning rates for distributions on a
smooth manifold, in a cost-sensitive setting. This generalises a classic result
of Audibert and Tsybakov. Building upon recent work of Chaudhuri and Dasgupta
we prove that these minimax rates are attained by the approximate nearest
neighbour algorithm, where neighbours are computed in a randomly projected
low-dimensional space. In addition, we give a bound on the number of dimensions
required for the projection which depends solely upon the reach and dimension
of the manifold, combined with the regularity of the marginal.
