Manifold Estimation and Singular Deconvolution Under Hausdorff Loss

We find lower and upper bounds for the risk of estimating a man-ifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure. 1. Introduction. Manifold learning is an area of intense research activity in machine learning and statistics. Yet a very basic question about manifold learning is still open, namely, how well can we estimate a manifold from n noisy samples? In this paper we investigate this question under various assumptions. Suppose we observe a random sample Y 1 ,. .. , Y n ∈ R D that lies on or near a d-manifold M where d < D. The question we address is: what is the minimax risk under Hausdorff distance for estimating M ? Our main assumption is that M is a d-dimensional, smooth Riemannian submanifold in R D ; the precise conditions on M are given in Section 2. Let Q denote the distribution of Y i. We shall see that Q depends on several things, including the manifold M , a distribution G supported on M and a model for the noise. We consider three noise models. The first is the noiseless model in which Y 1 ,. .. , Y n is a random sample from G. The second is the clutter noise model, in which