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We study the approximate nearest neighbour method for cost-sensitiveclassification on low-dimensional manifolds embedded within a high-dimensionalfeature space. We determine the minimax learning rates for distributions on asmooth manifold, in a cost-sensitive setting. This generalises a classic resultof Audibert and Tsybakov. Building upon recent work of Chaudhuri and Dasguptawe prove that these minimax rates are attained by the approximate nearestneighbour algorithm, where neighbours are computed in a randomly projectedlow-dimensional space. In addition, we give a bound on the number of dimensionsrequired for the projection which depends solely upon the reach and dimensionof the manifold, combined with the regularity of the marginal.

Minimax rates for cost-sensitive learning on manifolds with approximate nearest neighbours