  We consider semi-supervised regression when the predictor variables are drawn
from an unknown manifold. A simple two step approach to this problem is to: (i)
estimate the manifold geodesic distance between any pair of points using both
the labeled and unlabeled instances; and (ii) apply a k nearest neighbor
regressor based on these distance estimates. We prove that given sufficiently
many unlabeled points, this simple method of geodesic kNN regression achieves
the optimal finite-sample minimax bound on the mean squared error, as if the
manifold were known. Furthermore, we show how this approach can be efficiently
implemented, requiring only O(k N log N) operations to estimate the regression
function at all N labeled and unlabeled points. We illustrate this approach on
two datasets with a manifold structure: indoor localization using WiFi
fingerprints and facial pose estimation. In both cases, geodesic kNN is more
accurate and much faster than the popular Laplacian eigenvector regressor.
