  We consider a problem of manifold estimation from noisy observations. Many
manifold learning procedures locally approximate a manifold by a weighted
average over a small neighborhood. However, in the presence of large noise, the
assigned weights become so corrupted that the averaged estimate shows very poor
performance. We suggest a structure-adaptive procedure, which simultaneously
reconstructs a smooth manifold and estimates projections of the point cloud
onto this manifold. The proposed approach iteratively refines the weights on
each step, using the structural information obtained at previous steps. After
several iterations, we obtain nearly "oracle" weights, so that the final
estimates are nearly efficient even in the presence of relatively large noise.
In our theoretical study, we establish tight lower and upper bounds proving
asymptotic optimality of the method for manifold estimation under the Hausdorff
loss, provided that the noise degrades to zero fast enough.
