A tree-based regressor that adapts to intrinsic dimension

We consider the problem of nonparametric regression  , consisting of learning an arbitrary mapping f:X→Yf:X→Y from a data set of (x,y)(x,y) pairs in which the y   values are corrupted by noise of mean zero. This statistical task is known to be subject to a severe curse of dimensionality: if X⊂RDX⊂RD, and if the only smoothness assumption on f is that it satisfies a Lipschitz condition, it is known that any estimator based on n   data points will have an error rate (risk) of Ω(n−2/(2+D))Ω(n−2/(2+D)). Here we present a tree-based regressor whose risk depends only on the doubling dimension of XX, not on D  . This notion of dimension generalizes two cases of contemporary interest: when XX is a low-dimensional manifold, and when XX is sparse. The tree is built using random hyperplanes as splitting criteria, building upon recent work of Dasgupta and Freund (2008) [5]; and we show that axis-parallel splits cannot achieve the same finite-sample rate of convergence.

► Analysis of high-dimensional regression using random partition of data (RPtree).
► RPtree regression requires no dimension reduction: adapts to intrinsic dimension.
► General notion of intrinsic dimension captures manifold dimension, sparsity.
► Model selection made cheaper by monitoring data diameters in cells.
► Contrasting analysis: adaptivity of common dyadic tree method.